Introduction

A rewarding method for studying classes of graphs (that admit loops) and their properties, has been provided by the connectednesses and disconnectednesses. The terminology used for these classes are suggested by the properties of the graphs in the specific class. For example, the class of connected graphs is an example of a connectedness and with it is associated a disconnectedness namely the class of all graphs for which the only edges can be loops. This correspondence between the two types of classes is a Galois correspondence. Moreover, for a given connectedness \( \mathcal {C}\) with associated disconnectedness \(\mathcal {D} \), any graph has a decomposition (partition) into maximal components, which, when regarded as subgraphs, are graphs in the connectedness \(\mathcal {C}\) and such that the corresponding quotient graph is a graph in the disconnectedness \(\mathcal {D} . \) The theory of connectednesses and disconnectednesses for graphs has been defined and developed by Fried and Wiegandt [5] who also showed that this theory is just the radical theory of graphs from a categorical viewpoint. Radical theory has its origins in algebra and is one of the most successful tools in describing the structure of algebraic objects; think, for example, of the Jacobson radical of a ring and the nice structural description of a ring with Jacobson radical the zero ideal. The recent definition and development of a theory of congruences for graphs (that admit loops) has confirmed and strengthened this connection between the radical theory of algebra and the connectednesses theory of graphs. In particular, it was shown that a connectedness (and disconnectedness) can be obtained from an intersection of congruences, i.e., as a so called a Hoehnke radical. Moreover, a connectedness can be characterized using congruences which is in complete harmony with the use of ideals to characterize their algebraic counterparts, the radical classes (see [3, 13] and [14]).

Here we will present a radical theory for graphs that do not admit loops. A congruence theory for such graphs was introduced by Broere et al. [2]. They showed that, as for universal algebra, congruences give rise to isomorphism theorems as well as the characterization of subdirect products in terms of the intersection of congruences. When loops are not allowed, there is a significant restriction on the possible number of homomorphisms on a graph. Here we will show that this has a degenerative impact on certain aspects of the radical theory for such graphs. In particular, the semisimple class of a Hoehnke radical coincides with the class of all graphs in this case. But in spite of this, there are non-trivial connectednesses and disconnectednesses for such graphs. It should be mentioned that the notion of a congruence for a graph with no loops has already appeared in Sabidussi [10], quoting and using results (unpublished) from his PhD student Fawcett [4]. They used a congruence in a limited sense as the kernel of a strong graph homomorphism and no general theory of congruences was mentioned or discussed. In particular, they showed that subdirect products of graphs can be described in terms of congruences and they applied these to formulate and prove a version of Birkhoff’s Theorem for graphs: every graph is a subdirect product of subdirectly irreducible graphs. In their context, a subdirectly irreducible graph is either a complete graph or an almost complete graph.

To define the Hoehnke radical for graphs that do not admit loops, we recall the definition and theory of congruences for such graphs from Broere, Heidema and Pretorius [2] in some detail in Sect. 2 below. We will need a version of Birkhoff’s Theorem which is valid in the class of graphs and the mappings under consideration here, and this is given in Sect. 3. The main result here is that any graph has a subdirect representation of complete graphs; the subdirectly irreducible graphs are the complete graphs. In the last section we develop a radical theory for these loopless graphs. And it turns out to be an interesting contribution to the general radical theory in a very natural, non-artificial class of mathematical objects. In particular, we have non-trivial connectednesses and disconnectednesses, but all Hoehnke radicals degenerate. This means a graph need not have a maximal disconnectedness (semisimple) image. Moreover, all connectednesses and disconnectednesses come as complementary pairs.

All graphs considered are undirected and without multiple edges. Graphs have non-empty vertex sets with no loops, but edge sets may be empty. The vertex set V and edge set E of a graph G will typically be denoted by \(G=(V_{G},E_{G})\) with the subscripts when necessary. An edge in a graph will be denoted by the unordered pair xy for \(x,y\in V_{G},x\ne y\). A homomorphism \(f :G \rightarrow H\) from the graph G to the graph H is an edge preserving mapping from \(V_{G}\) into \(V_{H}\). Note that if \(ab\in E_{G} \), then \(f(a)\ne f(b)\) for a homomorphism f. A strong homomorphism is a homomorphism that sends “no edges” to “no edges” and if it is also a bijection, it is called an isomorphism. We use \(\cong \) to denote two isomorphic graphs. For a graph \(G=(V_{G},E_{G}) \), a subgraph \(H=(V_{H},E_{H})\) of G is a graph with \(V_{H}\subseteq V_{G}\) and \(E_{H}\subseteq \{ab\mid a,b\in H\}\cap E_{G}\). When \(E_{H}=\{ab\mid a,b\in H\}\cap E_{G} \), then H is called an induced subgraph of G. For a graph G, \(K_{G}\) will be the set \(K_{G}:=\{ab\mid a,b\in V_{G},a\ne b\}\) of all possible edges on G.

Congruences on graphs

A congruence on a graph G [2] is a pair \(\theta =(\sim ,\mathcal {E})\) such that

  1. (i)

    \(\sim \) is an equivalence relation on V;

  2. (ii)

    \(\mathcal {E}\) is a set of unordered pairs of different elements from \( V_{G}\) with \(E_{G}\subseteq \mathcal {E\subseteq }K_{G}\);

  3. (iii)

    when \(x\sim y \), then \(xy\not \in \mathcal {E};\) and

  4. (iv)

    (Substitution Property of \(\mathcal {E}\) with respect to \(\sim ) \) when \(x,y,x^{\prime },y^{\prime }\in V_{G}\), \(x\sim x^{\prime }\), \(y\sim y^{\prime }\), and \(xy\in \mathcal {E}\), then \(x^{\prime }y^{\prime }\in \mathcal {E}\).

A strong congruence on G is a pair \(\theta =(\sim ,\mathcal {E})\) where \(\sim \) is an equivalence relation on \(V_{G} \), \(\mathcal {E}=\{xy\mid x,y\in V_{G}\) and there are \(x^{\prime },y^{\prime }\in V_{G}\) with \(x\sim x^{\prime },y\sim y^{\prime }\) and \(x^{\prime }y^{\prime }\in E_{G}\}\) and condition (iii) is fulfilled.

Requirement (iii) ensures that the equivalence classes \( [x]:=\{y\in V_{G}\mid x\sim y\}\) are independent sets of vertices with respect to \(\mathcal {E}\), i.e., if \(a,b\in [x] \), then \(ab\notin E_{G}\) . It can easily be verified that a strong congruence is also a congruence. For a congruence \(\theta =(\sim ,\mathcal {E}) \), \(\mathcal {E}\) is called the congruence edge set. Congruences are partially ordered by the relation “contained in”: for two congruences \(\alpha =(\sim _{\alpha }, \mathcal {E}_{\alpha })\) and \(\beta =(\sim _{\beta },\mathcal {E}_{\beta })\) on G, \(\alpha \) is contained in \(\beta \), written as \(\alpha \subseteq \beta \), if \(\sim _{\alpha }\subseteq \sim _{\beta }\) and \( \mathcal {E}_{\alpha }\subseteq \mathcal {E}_{\beta }\). We will always use \( \Bumpeq \) to denote the identity relation (diagonal) on \(V_{G}\) (or any other set). The congruence \(\iota _{G}:=(\Bumpeq ,E_{G}) \), called the identity congruence on G, is the smallest congruence on G. It is a strong congruence. If G is a complete graph, i.e., \(E_{G}=K_{G}\), then G can have only one congruence namely the identity congruence \(\iota _{G}\) (see the proof of Proposition 3.2 below).

Given any homomorphism \( f :G \longrightarrow H\), a congruence on G, called the kernel of f and written as \(\ker f=(\sim _{f},\mathcal {E}_{f}) \), is defined by \(\sim _{f}=\{(x,y)\mid x,y\in V_{G},f(x)=f(y)\}\) and \(\mathcal {E}_{f}=\{uv\mid u,v\in V_{G},f(u)f(v)\in E_{H}\}\). It is immediately clear that \(\ker f\) is a congruence on G. With f is also associated the strong kernel of f written as \({\text {sker}} f=(\sim _{f},\mathcal {E}_{sf})\) with the same equivalence relation but \(\mathcal {E}_{sf}=\{xy\mid x,y\in V_{G}\) and there are \(x^{\prime },y^{\prime }\in V_{G}\) with \(x\sim _{f}x^{\prime }\), \( y\sim _{f}y^{\prime }\) and \(x^{\prime }y^{\prime }\in E_{G}\}\). This is a strong congruence on G and \({\text {sker}} f\subseteq \ker f;\) in fact, if \( \theta =(\sim _{f},\mathcal {E})\) is any congruence on G for some \(\mathcal { E}\), then \({\text {sker}} f\subseteq \theta \). If f is a strong homomorphism, then \(\ker f= {\text {sker}} f\). Note that a homomorphism f is injective if and only if \(\sim _{f}=\) \(\Bumpeq \). Moreover, if f is a surjective strong homomorphism, then f is an isomorphism if and only if \( \ker f=\iota _{G}.\)

Given any congruence \(\theta =(\sim ,\mathcal {E })\) on a graph \(G=(V_{G},E_{G})\), a new graph, denoted by \(G/\theta =(V_{G/\theta },E_{G/\theta })\) and called the quotient of G by \(\theta \), is defined by taking \(V_{G/\theta }:=\{[x]\mid x\in V_{G}\}\) and \(E_{G/\theta }:=\{[x][y]\mid xy\in \mathcal {E}\}\). The natural (canonical) mapping \(p_{\theta } :G \rightarrow G/\theta \) given by \(p_{\theta }(x)=[x]\) is a surjective homomorphism with \(\ker p_{\theta }=\theta \). In particular, for \(\theta =\iota _{G}\) we have \(G/\iota _{G}\) isomorphic to G. If \(\theta \) is a strong congruence, then \(p_{\theta }\) is a strong homomorphism with \(\ker p_{\theta }=\theta = {\text {sker}} p_{\theta }\). Suppose \(\theta =(\Bumpeq ,\mathcal {E})\) is a congruence on G for some suitable \(\mathcal {E.}\) Then \(G/\theta \) is the graph with vertex set \( V_{G/\theta }=V_{G}\) and edge set \(E_{G/\theta }=\mathcal {E}\) (here we identify \([x]=\{x\}\) with x).

For a given graph G, we denote the set of all congruences on G by \({\text {Con}}(G)\). We already know that \( {\text {Con}}(G)\) is a partially ordered set with respect to containment \(\subseteq .\) But we can say more. Any collection of congruences \(\{\theta _{i}=(\sim _{i}, \mathcal {E}_{i})\mid i\in I\}\subseteq {\text {Con}}(G)\) has a greatest lower bound in \({\text {Con}}(G)\) given by \(\bigcap \nolimits _{i\in I}\theta _{i}=(\sim ,\mathcal {E})\) where \(a\sim b\Leftrightarrow a\sim _{i}b\) for all \(i\in I\) and \(ab\in \mathcal {E}\Leftrightarrow ab\in \mathcal {E}_{i}\) for all \(i\in I\). This ensures that \({\text {Con}}(G)\) is a complete meet-semilattice with the meet given by the intersection defined above.

Let \(f :G \rightarrow H\) be a homomorphism and \(\theta =(\sim ,\mathcal {E})\) a congruence on G. Then \( f(\theta )\) means the pair \((f(\sim ),f(\mathcal {E}))\) with \(f(\sim ):=\{(f(a),f(b))\mid a,b\in V_{G},a\sim b\}\subseteq V_{H}\times V_{H}\) and \( f(\mathcal {E}):=\{f(a)f(b)\mid ab\in \mathcal {E\}}\subseteq \{xy\mid x,y\in V_{H}\}\). Note that \(f(\theta )\) need not be a congruence on the graph H,  but for a congruence \(\beta =(\sim _{\beta },\mathcal {E}_{\beta })\) on H,  we will compare \(f(\theta )\) with \(\beta \) in the usual sense: \(f(\theta )\subseteq \beta \) if \(f(\sim )\subseteq \sim _{\beta }\) and \(f(\mathcal {E} )\subseteq \mathcal {E}_{\beta }\).

When dealing with radicals, the basic tools are the appropriate versions of the algebraic isomorphism theorems for graph congruences. These theorems are discussed in detail in Broere, Heidema and Pretorius [2], but summarized below for ease of reference. We start with two auxiliary results.

Proposition 2.1

Let \(f :G \rightarrow H\) be a homomorphism. Then \(f(\ker f)\subseteq \iota _{H} \). If \(\alpha \) is a congruence on G with \(f(\alpha )\subseteq \iota _{H} \), or if \(\alpha \) is a strong congruence on G with \(f([x]_{\alpha })=\{f(x)\} \) for all \(x\in G \), then \(\alpha \subseteq \ker f.\)

Proof

We only show the last part. Let \(\ker f=(\sim _{f},\mathcal {E}_{f})\) and let \(\alpha =(\sim _{\alpha },\mathcal {E}_{\alpha })\) be a strong congruence on G with \(f([x]_{\alpha })=\{f(x)\}\) for all \(x\in G\). For \(a,b\in G \), if \( a\sim _{\alpha }b \), then \(a\in [b]_{\alpha }\) which gives \(f(a)\in f([b]_{\alpha })=\{f(b)\};\) i.e., \(f(a)=f(b)\) and so \(a\sim _{f}b\). Let \( ab\in \mathcal {E}_{\alpha }\). Since \(\alpha \) is strong, choose \(st\in [a]_{\alpha }[b]_{\alpha }\cap E_{G} \), \(s\in [a]_{\alpha }\), \( t\in [b]_{\alpha }\). Then \(f(a)f(b)=f(s)f(t)\in E_{H}\) and hence \( ab\in \mathcal {E}_{f}\). Thus \(\alpha \subseteq \) \(\ker f.\) \(\square \)

Proposition 2.2

Let \(f :G \rightarrow H\) and \(g :G \rightarrow K\) be surjective homomorphisms. Then \(\ker f\subseteq \ker g\) if and only if there is a homomorphism \( h :H \rightarrow K\) such that \(h\circ f=g.\)

Proof

Suppose \(\ker f=(\sim _{f},\mathcal {E}_{f})\) \(\subseteq \ker g=(\sim _{g}, \mathcal {E}_{g})\), i.e., \(\sim _{f}\subseteq \) \(\sim _{g}\) and \(\mathcal {E} _{f}\subseteq \mathcal {E}_{g}\). Define \(h :H \rightarrow K\) by \(h(y)=g(x)\) where \(x\in V_{G}\) with \(f(x)=y\). This map is well-defined, for if \( f(x^{\prime })=y=f(x) \), then \(x\sim _{f}x^{\prime }\) and so \(x\sim _{g}x^{\prime }\) which gives \(g(x)=g(x^{\prime })\). Clearly \(h\circ f=g.\) Next it is shown that h preserves edges: Let \(ab\in E_{H}\). By the surjectivity of f, there are \(a^{\prime },b^{\prime }\in V_{G}\) with \( f(a^{\prime })=a\) and \(f(b^{\prime })=b\). Thus \(f(a^{\prime })f(b^{\prime })\in E_{H}\) and so \(a^{\prime }b^{\prime }\in \mathcal {E}_{f}\subseteq \mathcal {E}_{g}\) which gives \(g(a^{\prime })g(b^{\prime })\in E_{K} \), i.e., \( h(a)h(b)\in E_{K}.\)

Conversely, suppose the homomorphism h with \(h\circ f=g\) is given. If \( f(a)=f(b) \), then \(g(a)=h(f(a))=h(f(b))=g(b)\) giving \(\sim _{f}\subseteq \) \( \sim _{g}\). Let \(ab\in \mathcal {E}_{f}\). Then \(f(a)f(b)\in E_{H}\) and hence \( h(f(a))h(f(b))\in E_{K}\) which gives \(g(a)g(b)\in E_{K}\). Thus \(ab\in \mathcal {E}_{g}\) and \(\mathcal {E}_{f}\subseteq \mathcal {E}_{g}\) follows. \(\square \)

(1) (First Isomorphism Theorem) [2] Let \(f :G \rightarrow H\) be a surjective homomorphism. Then \(G/\ker f\) is isomorphic to H.

The Second Isomorphism has also been given in [2], but we will present it here using a different (and more suggestive) notation. Let G be a graph with induced subgraph H. Then a congruence \(\theta =(\sim , \mathcal {E})\) on G induces a congruence \(H\cap \theta =(\sim _{H},\mathcal { E}_{H})\) on H with \(\sim _{H}=\{(a,b)\mid a,b\in V_{H}\) and \(a\sim b\}\) and \(\mathcal {E}_{H}=\{ab\mid a,b\in V_{H}\) with \(ab\in \mathcal {E}\}\). The mapping \(f :H \rightarrow G/\theta \) defined by \(f(a)=[a]\) for all \(a\in H\) is a homomorphism with \(\ker f=H\cap \theta \). Now \(f(V_{H})\) is a set of vertices of \(G/\theta \) on which we form the induced subgraph of \(G/\theta \) , denoted by \((H+\theta )/\theta \). Then, by the First Isomorphism Theorem, we have:

(2) (Second Isomorphism Theorem) [2] Let H be an induced subgraph of a graph G. Let \(\theta \) be a congruence on G. Then \( H\cap \theta \) is a congruence on H and \(H/H\cap \theta \cong (H+\theta )/\theta \) where \((H+\theta )/\theta \) is the induced subgraph of \(G/\theta \) on the vertex set \(\{[a]\mid a\in V_{H}\}.\)

(3) (Third Isomorphism Theorem) [2] Let G be a graph with \(\theta _{1}\!=\!(\sim _{1},\mathcal {E}_{1})\) and \(\theta _{2}=(\sim _{2}, \mathcal {E}_{2})\) two congruences on G for which \(\theta _{1}\subseteq \theta _{2}\). Then \(\theta _{2}/\theta _{1}:=(\sim ,\mathcal {E})\) is a congruence on \(G/\theta _{1}\) where \([a]_{1}\sim [b]_{1}\Leftrightarrow a\sim _{2}b\) and \([a]_{1}[b]_{1}\in \mathcal {E} \Leftrightarrow ab\in \mathcal {E}_{2}\). Moreover, \((G/\theta _{1})/(\theta _{2}/\theta _{1})\) is isomorphic to \(G/\theta _{2}.\)

A related result often used is:

(4) Let G be a graph with \(\theta \) a fixed congruence on G. Any congruence \(\xi \) on the graph \(G/\theta \) is of the form \(\xi =\alpha /\theta \) for some congruence \(\alpha \) on G with \(\theta \subseteq \alpha \). Moreover, there is a one-to-one correspondence between \(\{\alpha \mid \alpha \) is a congruence on G with \(\theta \subseteq \alpha \}\) and \( {\text {Con}}(G/\theta )\) which preserves inclusions and intersections [2].

Products, subdirect products and Birkhoff’s theorem

For an index set I, let \(G_{i}=(V_{i},E_{i})\) be a graph for all \(i\in I\). The product \(\prod \nolimits _{i\in I}G_{i}\) of the graphs \(G_{i}\) is the graph \(\prod \nolimits _{i\in I}G_{i}=(\prod \nolimits _{i\in I}V_{i},E)\) where \( \prod \nolimits _{i\in I}V_{i}\) is just the usual Cartesian product of the sets \( V_{i}\) and \(E=\{fg\mid f,g\in \prod \nolimits _{i\in I}V_{i}\) with \(f(i)g(i)\in E_{i}\) for all \(i\in I\}\). For every \(j\in I \), the j-th projection \(\pi _{j}:\prod \nolimits _{i\in I}G_{i}\rightarrow G_{j}\) defined by \(\pi _{j}(f)=f(j)\) for all \(f\in \prod \nolimits _{i\in I}V_{i}\) is a surjective homomorphism. An induced subgraph H of \(\prod \nolimits _{i\in I}G_{i}\) is called a subdirect product of the graphs \(G_{i},i\in I \), provided the restriction of each projection \(\pi _{j}\) to H is surjective. As in universal algebra, subdirect products can be expressed in terms of congruences and quotients.

Theorem 3.1

[2] A graph G is a subdirect product of graphs \(G_{i}\), \(i\in I \), if and only if for every \(i\in I\) there are congruences \(\theta _{i}\) on G with \(G_{i}\) isomorphic to \(G/\theta _{i}\) and \(\bigcap \nolimits _{i\in I}\theta _{i}=\iota _{G}.\)

Any graph has a representation as a subdirect product of graphs. For example, for a graph G, let \(\theta _{1}=\iota _{G}=\theta _{2}\). Then \(\theta _{1}\cap \theta _{2}=\iota _{G}\) and so G is a subdirect product of \(G/\theta _{1}\cong G\) and \(G/\theta _{2}\cong G\). It is thus interesting to know which graphs have the property that whenever it is a subdirect product of graphs \(G_{i},i\in I \), then at least one of the \(G_{i}\)’s must be isomorphic to G. Such graphs are called subdirectly irreducible. In view of the theorem above, a graph G is subdirectly irreducible if and only if any intersection of congruences on G that is the identity congruence must already include one congruence that is the identity. A graph which is not subdirectly irreducible, is subdirectly reducible. The corresponding notions for algebra are important, especially in the context of the well-known theorem of Birkhoff [1]: every non-trivial algebra is a subdirect product of subdirectly irreducible algebras. This theorem has meaning and validity for many other mathematical structures as well. For example, every non-trivial topological space is a subdirect product of copies of the Sierpiński space and the two-element indiscrete space (see Proposition 2.3 in [11]), noting that both these two two-element spaces are subdirectly irreducible topological spaces. Birkhoff’s Theorem is also valid in the category of graphs that admit loops [12], where it states that any non-trivial graph is a subdirect product of subdirectly irreducible graphs. Here the subdirectly irreducible graphs are \(B_{4} \), \(B_{5} \), \(B_{6}\) and \( A_{3}\) where the \(B_{i}\)’s are two-vertex graphs with \(V_{B_{i}}=\{0,1\} \), \( E_{B_{4}}=\{00,11\} \), \(E_{B_{5}}=\{00,01\}\) and \(E_{B_{6}}=\{00,01,11\},\) and \(A_{3}\) is the three-vertex graph with \(V_{A_{3}}=\{0,1,2\}\) and \( E_{A_{3}}=\{00,11,22,01,21\}.\)

As mentioned in the Introduction, Sabidussi [10] and Fawcett [4] formulated and proved Birkhoff’s theorem in the category of graphs which do not admit loops and for which the morphisms are the strong homomorphisms. For them, a congruence is the kernel of a strong homomorphism, which means that they only consider what we call here a strong congruence. These results are no longer true in our more general setting. We will need a version of Birkhoff’s Theorem for our more general setting, and this is what we look at next. Recall, a graph \(G=(V_{G},E_{G})\) is complete if \(E_{G}=\) \(K_{G}\). An almost complete graph is a graph G that has all possible edges except one, i.e., \(E_{G}=K_{G}-\{xy\}\) for some \(x,y\in V_{G},x\ne y.\)

Proposition 3.2

A graph is subdirectly irreducible if and only if it is complete.

Proof

Let \(G=(V_{G},E_{G})\) be a complete graph. Let \(\theta =(\sim ,\mathcal {E})\) be a congruence on G. Then \(K_{G}=E_{G}\subseteq \mathcal {E\subseteq } K_{G} \) and thus \(\mathcal {E=}K_{G}=E_{G}\). Since \(a\sim b\) implies \( ab\notin \mathcal {E,}\) it follows that \(a\sim b\) if and only if \(a=b\) and so \(\theta =(\sim ,\mathcal {E})=(\Bumpeq ,E_{G})=\iota _{G}\). This means the only congruence possible on G is the identity congruence and G is subdirectly irreducible. Conversely, suppose G is a subdirectly irreducible graph. If G is not complete, suppose firstly it misses exactly one edge xy, \(x,y\in V_{G},x\ne y\). Let \(\alpha \) and \(\beta \) be the congruences on G given by \(\alpha =(\Bumpeq ,K_{G})\) and \(\beta =(\sim ,E_{G})\) where the equivalence classes of \(\sim \) are \([x]=\{x,y\}=[y]\) and \( [a]=\{a\}\) for all \(a\in V_{G}-\{x,y\}\). It can be checked that both \(\alpha \) and \(\beta \) are non-identity congruences on G with \(\alpha \cap \beta =\iota _{G}\). This contradicts the choice of G, so if G is not complete, it must have at least two distinct edges missing, say xy and pq, \( x,y,p,q\in V_{G}\), \(x\ne y \), \(p\ne q\). In this case, let \(\alpha \) and \( \beta \) be the congruences on G defined by \(\alpha =(\Bumpeq ,\mathcal {E} _{\alpha })\) and \(\beta =(\Bumpeq ,\mathcal {E}_{\beta })\) where \(\mathcal {E} _{\alpha }:=E_{G}\cup \{xy\}\) and \(\mathcal {E}_{\beta }:=E_{G}\cup \{pq\}.\) Again it can be checked that both \(\alpha \) and \(\beta \) are non-identity congruences on G with \(\alpha \cap \beta =\iota _{G}\). This contradiction with the choice of G shows that G must be complete. \(\square \)

We can now formulate and prove Birkhoff’s Theorem for our category of graphs: the objects are undirected graphs, multiple edges and loops are not allowed; the morphisms are the graph homomorphisms (edge-preserving maps).

Theorem 3.3

Every graph is a subdirect product of subdirectly irreducible graphs, i.e., every graph is a subdirect product of complete graphs.

Proof

Let G be a graph. If G is complete, in particular also if G is a one-vertex graph, then we are done. If G is an almost complete graph, say \( E_{G}=K_{G}-\{xy\}\) for some \(x,y\in V_{G},x\ne y \), let \(\alpha \) and \( \beta \) be the congruences on G given by \(\alpha =(\Bumpeq ,K_{G})\) and \( \beta =(\sim ,E_{G})\) where the equivalence classes of \(\sim \) are \( [x]=\{x,y\}=[y]\) and \([a]=\{a\}\) for all \(a\in V_{G}-\{x,y\}\). As was seen in the proof of the previous result, both \(\alpha \) and \(\beta \) are non-identity congruences on G with \(\alpha \cap \beta =\iota _{G}\). Thus G is a subdirect product of the two complete graphs \(G/\alpha \) and \(G/\beta \), which takes care of this choice for G. Suppose thus G is missing at least two distinct edges. For any \(xy\in K_{G}-E_{G} \), let \(\theta _{xy}\) be the congruence on G defined by \(\theta _{xy}=(\sim _{xy},\mathcal {E}_{xy})\) where \(\sim _{xy}\) is the equivalence on \(V_{G}\) with equivalence classes \( [x]=\{x,y\}=[y]\) and \([a]=\{a\}\) for all \(a\in V_{G}-\{x,y\}\) and \(\mathcal {E }_{xy}:=K_{G}-\{xy\}\). The quotient graph \(G/\theta _{xy}\) is a complete graph, so the proof will be completed by Theorem 3.1 if we can show \(\cap \{\theta _{xy}\mid xy\in K_{G}-E_{G}\}=\iota _{G}\). Since the cardinality of \(K_{G}-E_{G}\) is at least two, we have \(\cap \{\sim _{\theta _{xy}}\mid xy\in K_{G}-E_{G}\}=\) \(\Bumpeq \). Moreover, \(\cap \{\mathcal {E}_{xy}\mid xy\in K_{G}-E_{G}\}=E_{G}\). Thus \(\cap \{\theta _{xy}\mid xy\in K_{G}-E_{G}\}=\iota _{G}.\) \(\square \)

Graph radicals

The origins of radical theory goes back to the early twentieth century with the work of Wedderburn on finite dimensional algebras. This was extended to ring theory with Köthe’s nilradical, the Jacobson radical and subsequently many other radicals. These, together with some developments in group theory led to the axiomization of the radical concept by Kurosh and independently Amitsur for rings, groups and more generally omega-groups in the early fifties. In recognition of their contributions in establishing this abstract approach to radicals, these radicals are often called Kurosh–Amitsur radicals (KA-radicals for short). For a good overview of the radical theory of associative rings, Gardner and Wiegandt [6] can be consulted. In the sixties, Hoehnke [7] used congruences to define a radical for universal algebras; now known as a Hoehnke radical (H-radical). Then necessary and sufficient conditions to ensure that the Hoehnke radical becomes a Kurosh–Amitsur radical have been determined for algebraic structures (Mlitz [9]), and they can also be adopted for the non-algebraic structures ( for example, for graphs that allow loops, see [3]).

One of the characterizations of the radicals of rings can be interpreted in a general categorical setting. This led to the definition and development of a radical theory in many divergent mathematical structures. The most general approach to radical theory is to be found in Márki, Mlitz and Wiegandt [8]. Connectednesses and disconnectednesses for graphs that allow loops have been defined and developed by Fried and Wiegandt [5] showing that they are just the KA-radical and -semisimple classes respectively in this category. It was shown in [3] that they can be obtained as H-radicals given by the intersection of graph congruences. Further connections between the connectednesses and disconnectednesses of these graphs and their congruences were investigated in [13] and [14]. In all these cases KA-radicals are special instances of H-radicals.

Here we will define and develop a radical theory (= connectedness theory) for graphs that do not admit loops. The prohibition on loops imposes a significant restriction on the admissible maps and will consequently lead to a different theory. In fact, it will be shown that the radical theory in the category of graphs that do not admit loops is different to all existing radical theories. In particular, although all Hoehnke radicals are degenerative, there are non-trivial connectednesses (radical classes) and disconnectednesses (semisimple classes). Contrary to radical theories in other categories, this means that objects need not have maximal semisimple images. Moreover, all radical and semisimple classes come as complementary pairs. We start by defining a Hoehnke radical and give some of their salient features. But, as we shall see, all such radicals are trivial. Then we define the connectednesses and disconnectednesses, give non-trivial examples of such classes and show that they always come as complementary pairs.

Radical theory is usually defined and developed in a prescribed universal class. A class \(\mathcal {W}\) of graphs is called a universal class if it is non-empty, closed under homomorphic images and closed under the taking of subgraphs (= strongly hereditary). From the definition, it follows that \(\mathcal {W}\) contains a one-vertex graph, and consequently all one-vertex graphs. We identify all the one-vertex-graphs, denote them by T and call them the trivial graphs. For any graph G, the completion of G is the graph \(G^{c}\) with the same vertex set as G and edge set \(K_{G}\). Since \(G^{c}\) is a homomorphic image of G\(\mathcal {W}\) contains the completion of all graphs \(G\in \mathcal {W.}\) We will assume \(\mathcal {W}\) is non-trivial, i.e., it has at least one graph with two or more vertices. Since \(\mathcal {W}\) is strongly hereditary, it thus contains the two-vertex graph with no edges \(B_{1}\) and its completion \( B_{1}^{c}\). All considerations relating to the radicals of graphs will be inside the class \(\mathcal {W}\). As usual, we do not distinguish between isomorphic graphs.

Definition 4.1

An H-radical on \(\mathcal {W}\) is a function \(\varrho \) that assigns to every graph G in \(\mathcal {W}\) a congruence \(\varrho (G)=\varrho _{G}\) on G such that:

  1. (H1)

    if \(f :G \rightarrow H\) is a surjective homomorphism in \(\mathcal {W},\) then \(f(\varrho _{G})\subseteq \varrho _{H}\); and

  2. (H2)

    for all graphs \(G\in \mathcal {W} \), \(\varrho (G/\varrho _{G})=\iota _{G/\varrho _{G}}\), the identity congruence on \(G/\varrho _{G}\).

For an H-radical \(\varrho \), the class \(\mathcal {S}_{\varrho }=\{G\in \mathcal {W}\mid \varrho (G)=\iota _{G}\}\) is called the associated semisimple class and \(\mathcal {R}_{\varrho }=\{G\in \mathcal {W}\mid G/\varrho _{G}\) is the one-element graph} is the associated radical class. It can easily be checked that \(\mathcal {S}_{\varrho }\cap \mathcal {R} _{\varrho }=\{T\}\) and \(\mathcal {R}_{\varrho }\) is always homomorphically closed, i.e., if \(G\in \mathcal {R}_{\varrho }\) and \( f :G \rightarrow H\) is a surjective homomorphism, then \(H\in \mathcal {R} _{\varrho }\). The next result is really the essence of this radical and shows that a Hoehnke radical is very general and always is of a prescribed form. A class of graphs \(\mathcal {M}\) in \(\mathcal {W}\) is an abstract class provided it contains all the one vertex graphs in \(\mathcal { W}\) and it is closed under isomorphic copies. All subclasses of \(\mathcal {W}\) under consideration are assumed to be abstract, even though it may not always be stated explicitly. For a class \(\mathcal {M}\) in \(\mathcal {W} \), we use \(\overline{\mathcal {M}}\) to denote the subdirect closure of \( \mathcal {M} \), i.e., the class of all graphs that are subdirect products of graphs from \(\mathcal {M}\). Clearly \(\mathcal {M}\subseteq \overline{\mathcal {M }}\) and we say \(\mathcal {M}\) is subdirectly closed if \(\mathcal {M}= \overline{\mathcal {M}}.\)

Theorem 4.2

Let \(\varrho \) be a mapping that assigns to any graph G in \(\mathcal {W}\) a congruence \(\varrho (G)=\varrho _{G}\) on G. Then \(\varrho \) is an H -radical on \(\mathcal {W}\) if and only if there is an abstract class of graphs \(\mathcal {M}\) in \(\mathcal {W}\) such that for all G in \(\mathcal {W},\) \(\varrho (G)=\cap \{\theta \mid \theta \) is a congruence on G for which \( G/\theta \in \mathcal {M}\}\). Furthermore, \(\mathcal {S}_{\varrho }=\) \( \overline{\mathcal {M}}.\)

Proof

Let \(\varrho \) be an H-radical. Then the semisimple class \(\mathcal {S} _{\varrho }=\{G\in \mathcal {W}\mid \varrho (G)=\iota _{G}\}\) is an abstract class. We show that \(\varrho (G)=\varphi \) where \(\varphi :=\cap \{\theta \mid \theta \) is a congruence on G for which \(G/\theta \in \mathcal {S} _{\varrho }\}\). By (H2), \(\varrho (G/\varrho _{G})\in \mathcal {S}_{\varrho }\) and so \(\varphi \subseteq \varrho (G)\) since \(\varrho (G)\) is just one of these \(\theta \)’s. Let \(\theta \) be any congruence on G for which \( G/\theta \in \mathcal {S}_{\varrho }\). For the canonical quotient map \( p_{\theta } :G \rightarrow G/\theta \), by (H1) we have \(p_{\theta }(\varrho (G))\subseteq \varrho (p_{\theta }(G))=\varrho (G/\theta )=\iota _{G/\theta }\). This means \(\varrho (G)\subseteq \theta \), hence \(\varrho (G)\subseteq \varphi \) and \(\varrho (G)=\cap \{\theta \mid \theta \) is a congruence on G for which \(G/\theta \in \mathcal {S}_{\varrho }\}\) follows. We still have to show \(\overline{\mathcal {S}_{\varrho }}=\mathcal {S}_{\varrho }\). Suppose a graph G is a subdirect product of graphs \(G_{i}\in \mathcal {S}_{\varrho },\) \(i\in I\). By Theorem 3.1 we know there are congruences \(\alpha _{i}\) on G with \(G/\alpha _{i}=G_{i}\in \mathcal {S}_{\varrho }\) and \( \bigcap \nolimits _{i\in I}\alpha _{i}=\iota _{G}\). But then \(\varrho (G)=\cap \{\theta \mid \theta \) is a congruence on G for which \(G/\theta \in \mathcal {S}_{\varrho }\}\subseteq \bigcap \nolimits _{i\in I}\alpha _{i}=\iota _{G}\) and hence \(G\in \mathcal {S}_{\varrho }.\)

Conversely, suppose \(\mathcal {M}\) is some abstract class of graphs in \( \mathcal {W}\) and \(\varrho (G)=\cap \{\theta \mid \theta \) is a congruence on G for which \(G/\theta \in \mathcal {M}\}\) for all G in \(\mathcal {W.}\) For (H1), let \(\theta \) be a congruence on G and consider the canonical surjective homomorphism \(f :G \rightarrow G/\theta \). We show \(f(\varrho (G))\subseteq \varrho (f(G))=\varrho (G/\theta )\). Let \(\alpha \) be a congruence on \(G/\theta \) with \((G/\theta )/\alpha \in \mathcal {M}\). By the Third Isomorphism Theorem, \(\alpha \) is of the form \(\alpha =\beta /\theta \) for some congruence \(\beta \) on G with \(\theta \subseteq \beta \) and \( G/\beta \) is isomorphic to \((G/\theta )/\alpha \in \mathcal {M}\). Hence \( \varrho (G)\subseteq \beta \) and so \(f(\varrho (G))\subseteq \beta /\theta =\alpha \) from which it can be concluded that \(f(\varrho (G))\subseteq \varrho (G/\theta )\). For (H2), using the Third Isomorphism Theorem again, we have \(\varrho (G/\varrho _{G})=\cap \{\alpha \mid \alpha \) is a congruence on \(G/\varrho _{G}\) with \((G/\varrho _{G})/\alpha \in \mathcal {M} \}=\cap \{\beta /\varrho _{G}\mid \beta \) is a congruence on G, \(\varrho _{G}\subseteq \beta \) and \(G/\beta \cong (G/\varrho _{G})/(\beta /\varrho _{G})\in \mathcal {M}\}=\varrho _{G}/\varrho _{G}=\iota _{G/\varrho _{G}}\). Thus (H2) follows.

Lastly we show \(\mathcal {S}_{\varrho }=\) \(\overline{\mathcal {M}}\). For this we first observe that \(\mathcal {M\subseteq S}_{\varrho }\) since \(G/\iota _{G}\cong G\in \mathcal {M,}\) which gives \(\varrho (G)=\iota _{G}\). Then \( \overline{\mathcal {M}}\subseteq \overline{\mathcal {S}_{\varrho }}=\mathcal {S} _{\varrho } \), the last equality follows from the first part of the proof. If \(G\in \mathcal {S}_{\varrho } \), then \(\iota _{G}=\varrho (G)=\cap \{\theta \mid \theta \) is a congruence on G for which \(G/\theta \in \mathcal {M}\}\) which means G is a subdirect product of graphs from \(\mathcal {M}\) (Theorem 3.1) and hence in \(\overline{\mathcal {M}}.\) \(\square \)

In the context of the theorem above, we say that the H-radical \( \varrho \) is determined by the class \(\mathcal {M}\) if \(\mathcal {M}\subseteq \mathcal {S}_{\varrho }\) and \(\overline{\mathcal {M}}=\mathcal {S}_{\varrho }\); the class \(\mathcal {M}\) is not necessarily unique. The corollary below emphasizes the salient features of an H-radical.

Corollary 4.3

  1. (1)

    The semisimple class of any H-radical is subdirectly closed.

  2. (2)

    \(\varrho (G)\) is the smallest congruence on G for which \( G/\varrho (G)\) is semisimple (i.e., if \(\theta \) is a congruence on G with \(G/\theta \in \mathcal {S}_{\varrho } \), then \(\varrho (G)\subseteq \theta ).\) Or, equivalently, \(G/\varrho (G)\) is the largest semisimple image of G (in the following sense: if \(g :G \rightarrow H\) is a surjective homomorphism with \(H\in \mathcal {S}_{\varrho } \), then there is a homomorphism \(h :G /\varrho (G)\rightarrow H\) such that \(h\circ p=g\) where \(p :G \rightarrow G/\varrho (G)\) is the canonical quotient map).

  3. (3)

    For any abstract class of graphs \(\mathcal {M}\) in \(\mathcal {W}\) for which we have for every G in \(\mathcal {W}\) a congruence \(\theta \) on G with \(G/\theta \in \mathcal {M}\), define \(\varrho (G)=\cap \{\theta \mid \theta \in \mathcal {C}(G)\) with \(G/\theta \in \mathcal {M}\}\). Then \(\varrho \) is an H- radical and a graph is semisimple if and only if it is a subdirect product of graphs from \(\mathcal {M}.\)

Number (3) above is really the holy grail of radical theory. One would like to have a nice, well-behaved class of graphs \(\mathcal {M}\) and then, if a graph is semisimple with respect to the corresponding radical, it is a subdirect product of graphs from the class \(\mathcal {M}\). Properties of the class or of the graphs inside the class \(\mathcal {M}\) may lead to stronger representations as is often seen, for example, in the radical theory of rings. In categories where congruences can be identified by a distinguished subobject (e.g. for rings, a congruence is completely determined by an ideal, the congruence class of the additive identity), any class \(\mathcal {M}\) of objects will fulfil the requirement of (3) above, but this is not true in general. Fortunately we can say exactly when it will be in our case. Let \(\mathcal {K}_{\mathcal {W}}\) be the set of all complete graphs in \(\mathcal {W}.\)

Lemma 4.4

Let \(\mathcal {M}\) be an abstract class of graphs in \(\mathcal {W}\). Then for every G in \(\mathcal {W}\) there is a congruence \(\theta \) on G with \( G/\theta \in \mathcal {M}\) if and only if \(\mathcal {K}_{\mathcal {W}}\subseteq \mathcal {M}.\)

Proof

Suppose the condition holds and let \(G\in \mathcal {K}_{\mathcal {W}}\). By the assumption, we know there is a congruence \(\theta =(\sim ,\mathcal {E})\) on G with \(G/\theta \in \mathcal {M}\). But the only congruence on a complete graph is the identity congruence. Thus \(\theta =\iota _{G}\) and we conclude \( G\cong G/\iota _{G}\in \mathcal {M}\). Conversely, suppose \(\mathcal {K}_{ \mathcal {W}}\subseteq \mathcal {M}\). For any graph \(G\in \mathcal {W,}\) we have the canonical bijective homomorphism \(f :G \rightarrow G^{c}\) from G to its completion \(G^{c}\) with \(G/\ker f\cong G^{c}\in \mathcal {K}_{\mathcal {W} }\subseteq \mathcal {M}\). \(\square \)

Theorem 4.5

Let \(\varrho \) be an H-radical on \(\mathcal {W.}\) Then \(\mathcal {S} _{\varrho }=\mathcal {W} \), i.e., \(\varrho (G)=\iota _{G}\) for all \(G\in \mathcal {W} \), and \(\mathcal {R}_{\varrho }=\{T\}.\)

Proof

By the Lemma, we have \(\mathcal {K}_{\mathcal {W}}\subseteq \mathcal {S} _{\varrho }\). For any graph \(G\in \mathcal {W} \), we know by Theorem 3.3 that G is a subdirect product of complete graphs \(G_{i},i\in I\) for some index set I. Each \(G_{i}\) is a homomorphic image of \(G\in \mathcal {W}\) and hence also in \(\mathcal {W}\). This gives \(G_{i}\in \mathcal {K}_{\mathcal {W} }\subseteq \mathcal {S}_{\varrho }\) for all i. By Corollary 4.3, we know \( \mathcal {S}_{\varrho }\) is closed under subdirect products, hence \(G\in \mathcal {S}_{\varrho }\). Thus \(\mathcal {S}_{\varrho }=\mathcal {W.}\) Let \( G\in \) \(\mathcal {R}_{\varrho }\). Since \(\mathcal {R}_{\varrho }\) is homomorphically closed, \(G^{c}\in \mathcal {R}_{\varrho }\cap \mathcal {K}_{ \mathcal {W}}\subseteq \mathcal {R}_{\varrho }\cap \mathcal {S}_{\varrho }=\{T\}\). Thus \(G=G^{c}=T.\) \(\square \)

Any H-radical on a universal class of graphs that do not admit loops is thus degenerative (\(\varrho (G)=\iota _{G}\) for all G). On the other hand, in general radical theory, the semisimple objects are usually regarded as well-behaved and sought after objects. One interpretation of this equality \(\mathcal {S}_{\varrho }=\mathcal {W,}\) is then that all the graphs that do not admit loops are good graphs! Next we define the connectednesses and disconnectednesses in the universal class \(\mathcal {W}.\)

Kurosh–Amitsur radicals Here we define and look at some of the properties of a connectedness (= KA-radical class) and a disconnectedness (= KA-semisimple class) in the universal class \(\mathcal {W}\) of graphs. A class \(\mathcal {C}\subseteq \mathcal {W}\) is a connectedness if it satisfies the following condition: A graph G is in \( \mathcal {C}\) if and only if every non-trivial homomorphic image of G has a non-trivial induced subgraph which is in \(\mathcal {C}\). A class \(\mathcal {D} \subseteq \mathcal {W}\) is a disconnectedness if it satisfies the condition: A graph G is in \(\mathcal {D}\) if and only if every non-trivial induced subgraph of G has a non-trivial homomorphic image which is in \( \mathcal {D}\). The trivial graph T is always in any connectedness and also in any disconnectedness. If \(\mathcal {M}\subseteq \mathcal {W}\) is a hereditary class, then \(\mathcal{UM}\mathcal{}:=\{G\in \mathcal {W}\mid G\) has no non-trivial homomorphic image in \(\mathcal {M}\}\) is a connectedness and if \( \mathcal {H}\subseteq \mathcal {W}\) is a homomorphically closed class, then \( \mathcal{SH}\mathcal{}:=\{G\in \mathcal {W}\mid G\) has no non-trivial induced subgraph in \(\mathcal {H}\}\) is a disconnectedness.

Proposition 4.6

Any connectedness is homomorphically closed and any disconnectedness is strongly hereditary.

Proof

We only check the strong hereditariness of a disconnectedness \(\mathcal {D}\). Firstly we show \(\mathcal {D}\) is hereditary. Let H be an induced subgraph of \(G\in \mathcal {D}\). To show \(H\in \mathcal {D} \), consider any non-trivial induced subgraph S of H. Then S is a non-trivial induced subgraph of \( G\in \mathcal {D}\) and by definition of a disconnectedness, S has a non-trivial homomorphic image in \(\mathcal {D}\). Thus \(H\in \mathcal {D.}\) Next we show that \(\mathcal {D}\) is strongly hereditary. Let H be a subgraph of \(G\in \mathcal {D}\). If H is trivial, we are done, so suppose \( H\ne T\). Let L be a non-trivial induced subgraph of H. Let \(\overline{L} \) be the induced subgraph of G on \(V_{L}\). We know from the first part that \(\overline{L}\in \mathcal {D}\). Thus \(\overline{L}\) is a non-trivial homomorphic image of L which is in \(\mathcal {D}\). Hence \(H\in \mathcal {D.}\) \(\square \)

From the preceding, we thus have: If \(\mathcal {C}\) is a connectedness, then \(\mathcal {D}:=\mathcal{SC}\mathcal{}\) is a disconnectedness, called the disconnectedness corresponding to \(\mathcal {C} \), and if \( \mathcal {D}\) is a disconnectedness, then \(\mathcal {C}:=\mathcal{UD}\mathcal{}\) is a connectedness, called the connectedness corresponding to \(\mathcal {D }\). Moreover, it can be shown that a class \(\mathcal {C}\subseteq \mathcal {W}\) is a connectedness if and only if \(\mathcal {C}=\mathcal {USC}\) and a class \( \mathcal {D}\subseteq \mathcal {W}\) is a disconnectedness if and only if \( \mathcal {D}=\mathcal {SUD}.\)

Proposition 4.7

Any disconnectedness is closed under subdirect products.

Proof

Let \(\mathcal {D}\) be a disconnectedness and suppose the graph G is a subdirect product of the graphs \(G_{i}\in \mathcal {D}\), \(i\in I\). This means there are congruences \(\theta _{i}\) on G such that \(G/\theta _{i}\cong G_{i}\) for each \(i\in I\) and \(\bigcap \nolimits _{i}\theta _{i}=\iota _{G}\). We show \(G\in \mathcal {D}\). Let H be a non-trivial induced subgraph of G. For every i, \(H\cap \theta _{i}\) is a congruence on H and by the Second Isomorphism Theorem, \(H/(H\cap \theta _{i})\cong (H+\theta _{i})/\theta _{i}\) which is an induced subgraph of \(G/\theta _{i}\in \mathcal {D}\). Hence \( H/(H\cap \theta _{i})\in \mathcal {D}\) and it is a homomorphic image of H for every i. At least one of them is non-trivial. Indeed, if \(H/(H\cap \theta _{i})\cong T\) for all i, then for any \(a,b\in V_{H}\) we have a is \((H\cap \theta _{i})\)-equivalent to b for all i. This means a is \( (\bigcap \nolimits _{i}\theta _{i})\)-equivalent to b. But \(\bigcap \nolimits _{i} \theta _{i}=\iota _{G}\) and so we get \(a=b\) for all \(a,b\in V_{H}\). This contradicts H being non-trivial. Thus H has at least one non-trivial homomorphic image \(H/(H\cap \theta _{j})\in \mathcal {D}\) for some \(j\in I\) and we conclude that \(G\in \mathcal {D}.\) \(\square \)

By Theorem 4.5 we immediately have:

Proposition 4.8

Let \(\mathcal {D}\) be a disconnectedness. Suppose there is an H-radical \( \varrho \) such that its semisimple class \(\mathcal {S}_{\varrho }=\mathcal {D} . \) Then \(\mathcal {D}=\mathcal {W}\) and the corresponding connectedness \( \mathcal {C}=\mathcal{UD}\mathcal{}\) is \(\mathcal {C}=\{T\}.\)

The significance of this result follows from the following feature of all known radical theories. For a radical theory in a given universal class \(\mathcal {W}\) of objects and a semisimple class \(\mathcal {D}\) in \( \mathcal {W} \), any object in \(\mathcal {W}\) has a maximal semisimple image in the following sense. For any \(A\in \mathcal {W} \), there is a surjective morphism \(q:A\rightarrow A_{\mathcal {D}}\) with \(A_{\mathcal {D}}\in \mathcal {D }\) such that for any surjective morphism \(f:A\rightarrow D\) with \(D\in \mathcal {D} \), there is a morphism \(g:A_{\mathcal {D}}\rightarrow D\) for which \(g\circ q=f\). If there is a congruence theory available in \(\mathcal {W},\) then this means that every \(A\in \mathcal {W}\) has a congruence \(\delta _{A}\) with \(A/\delta _{A}\cong A_{\mathcal {D}}\) and for every congruence \(\theta \) on A for which \(A/\theta \in \mathcal {D} \), we have \(\theta \subseteq \delta _{A}\). Thus we have an H-radical \(\delta \) on \(\mathcal {W}\) with \( \delta (A)=\delta _{A}\) for all \(A\in \mathcal {W},\mathcal {S}_{\delta }= \mathcal {D}\) and \(\mathcal {R}_{\delta }=\mathcal{UD}\mathcal{}\). This raises the question whether there are any non-trivial connectednesses and disconnectednesses in the universal class \(\mathcal {W}\) of loopless graphs. By example we will show that there are and we have chosen a very special example. For a connectedness \(\mathcal {C}\) with corresponding disconnectedness \(\mathcal {D}\) in \(\mathcal {W,}\) we know \(\mathcal {C}\cap \mathcal {D}=\{T\}\) and the pair \((\mathcal {C},\mathcal {D})\) is called complementary if \(\mathcal {C}\cup \mathcal {D}=\mathcal {W}\). Of course, there are two trivial such pairs, namely \((\mathcal {W},\{T\})\) and \( (\{T\},\mathcal {W})\). The existence of non-trivial complementary pairs was first observed in the category of graphs that allow loops by Fried and Wiegandt [5]. Such pairs can also be found in the radical theory for \( \mathcal {S}\)-acts, see Wiegandt [16], with a general condition for their existence given in [15] (which however, is not applicable here).

Example 4.9

Suppose \(\mathcal {K,}\) the class of all complete graphs, is contained in \( \mathcal {W}\). The class \(\mathcal {K}\) is homomorphically closed, hence \( \mathcal {D}:=\mathcal{SK}\mathcal{}\) is a disconnectedness and the corresponding connectedness is \(\mathcal {C}=\mathcal {USK}\). It follows that \(\mathcal {D} =\{G\in \mathcal {W}\mid E_{G}=\emptyset \}\) and \(\mathcal {C}=\{G\in \mathcal { W}\mid G=T\) or \(E_{G}\ne \emptyset \}\) which is clearly a complementary pair. It is a non-trivial complementary pair since \(\mathcal {W}\) contains all the complete graphs and is strongly hereditary (so \(\mathcal {D}\ne \{T\})\). Moreover, no graph G in \(\mathcal {W}\) with \(E_{G}\ne \emptyset \) has a homomorphic image in \(\mathcal {D}\) and thus certainly not a maximal one. \(\mathcal {D}\) is not a connectedness (not homomorphically closed) and \( \mathcal {C}\) is not a disconnectedness (not strongly hereditary). Any graph \( G\in \mathcal {W}\) has a homomorphic image in \(\mathcal {C} \), namely \( G\rightarrow G^{c}\in \mathcal {K}\subseteq \mathcal {C}\) where \(G^{c}\) is the completion of G and it is maximal in the sense that if H is any other complete homomorphic image of G, then \(H\subseteq G^{c}.\)

This example brings a rather special feature of the general radical theory to the fore: here we have a very natural universal class with a radical theory in which all the Hoehnke radicals degenerate, but there are non-trivial KA-radicals and they do not come from Hoehnke radicals. But there are more surprises; the example above is not really special at all. Recall, \(B_{1}\) is the graph with two vertices and no edge. For any connectedness \(\mathcal {C}\) with corresponding disconnectedness \(\mathcal {D} , \) we must always have \(B_{1}\in \mathcal {C}\) or \(B_{1}\in \mathcal {D}\). By \(K_{n}\) we denote the complete graph on n vertices, \(n\ge 1\). Then \( K_{2}=B_{1}^{c}\) and \(K_{1}=T.\)

Proposition 4.10

Let \(\mathcal {C}\subseteq \mathcal {W}\) be a connectedness with corresponding disconnectedness \(\mathcal {D}=\mathcal{SC}\mathcal{}\). Then \((\mathcal {C},\mathcal {D})\) is a complementary pair. If \(B_{1}\in \mathcal {C} \), then the pair \((\mathcal { C},\mathcal {D})=(\mathcal {W},\{T\})\) is trivial, and if \(B_{1}\in \mathcal {D} \), then the pair \((\mathcal {C},\mathcal {D})\) need not be trivial.

Proof

Suppose \(B_{1}\in \mathcal {C.}\) Since \(\mathcal {C}\) is homomorphically closed, also \(K_{2}\) is in \(\mathcal {C}\). This means \(E_{G}=\emptyset \) for all \(G\in \mathcal {D}\). Indeed, if there is a graph G in \(\mathcal {D}\) with \(E_{G}\ne \emptyset \), then \(K_{2}\) is an induced subgraph of G and the hereditariness of \(\mathcal {D}\) gives the contradiction \(K_{2}\in \mathcal {C}\cap \mathcal {D}=\{T\}\). From this we get \(\mathcal {D}=\{T\},\) for if \(G\in \mathcal {D}\) and G has more than one vertex, then \(B_{1}\) is an induced subgraph of G and again the hereditariness of \(\mathcal {D}\) gives the contradiction \(B_{1}\in \mathcal {C}\cap \mathcal {D}=\{T\}\). Let G be any non-trivial graph in \(\mathcal {W}\). We show that G is in \(\mathcal {C }\) by the definition of a connectedness. Let H be any non-trivial homomorphic image of G. Then at least one of \(B_{1}\) or \(K_{2}\) is a (non-trivial) induced subgraph of H and, since both \(B_{1}\) and \(K_{2}\) are in \(\mathcal {C} \), we can conclude that \(G\in \mathcal {C}.\)

Suppose now \(B_{1}\notin \mathcal {C,}\) i.e., \(B_{1}\in \mathcal {D} . \) Then \(E_{G}\ne \emptyset \) for all non-trivial graphs G in \(\mathcal {C }\) since any non-trivial graph which has no edges can be mapped onto \(B_{1}.\) We show \(G\in \mathcal {C}\cup \mathcal {D}\). Let \(G\in \mathcal {W}\). If \(G\in \mathcal {D}=\mathcal{SC}\mathcal{}\) we are done; suppose thus \(G\notin \mathcal {SC.}\) This means G has a non-trivial induced subgraph H which is in \(\mathcal {C } \), and hence \(E_{H}\ne \emptyset \). We show that \(G\in \mathcal {C,}\) again by using the definition of a connectedness. Let L be a non-trivial homomorphic image of G, say \(f :G \rightarrow L\). Now f(H) is non-trivial since \(E_{H}\ne \emptyset \). Moreover, \(f(H)\in \mathcal {C}\) and f(H) is a subgraph of L. Let \(\overline{f(H)}\) be the induced subgraph of L on \( V_{f(H)}\). Since \(\overline{f(H)}\) is a homomorphic image of f(H) , we may conclude that \(\overline{f(H)}\) is a non-trivial induced subgraph of L which is in \(\mathcal {C}\). Hence \(G\in \mathcal {C}\) and we are done. The second statement follows from the preceding example. \(\square \)

This result gives another special feature of the radical theory in this universal class \(\mathcal {W}\): any connectedness (respt. disconnectedness) is a connectedness (respt. disconnectedness) in a complementary pair. We conclude with more examples of connectednesses and disconnectednesses (which includes Example 4.9 as a special case).

Example 4.11

Suppose \(K_{n}\in \mathcal {W}\) for \(n=1,2,3,\ldots \). For each \(n\ge 1 \), let \( \mathcal {C}_{n}:=\{G\in \mathcal {W}\mid G=T\) or if \(G\ne T \), then \(K_{n}\) is an induced subgraph of \(G\}\) and let \(\mathcal {D}_{n}:=\{G\in \mathcal {W} \mid G=T\) or if \(G\ne T \), then \(K_{n}\) is not an induced subgraph of \(G\}.\) Using the definitions, it is straightforward to check that \(\mathcal {C}_{n}\) and \(\mathcal {D}_{n}\) form a complementary pair of connectednesses and disconnectednesses. Clearly \(\mathcal {C}_{1}=\mathcal {W}\) and it can be checked that \(\mathcal {C}_{2}=\{G\in \mathcal {W}\mid G=T\) or \(E_{G}\ne \emptyset \}\). Let \(\mathcal {C}\) be any connectednesss in \(\mathcal {W}\). As we have seen in the proof of Proposition 4.10, if \(B_{1}\in \mathcal {C},\) then \(\mathcal {C}=\mathcal {W}=\mathcal {C}_{1}\). Suppose thus \(B_{1}\notin \mathcal {C}\). Again from the proof of Proposition 4.10 we see that \(\mathcal { C}\subseteq \mathcal {C}_{2}\). This means \(\mathcal {W}\) has a largest proper connectedness \(\mathcal {C}_{2}\) and thus a smallest non-trivial disconnectedness \(\mathcal {D}_{2}=\{G\in \mathcal {W}\mid E_{G}=\emptyset \}.\) If \(\mathcal {C}\) contains a graph with n vertices, \(n>1 \), then \(K_{n}\in \mathcal {C}\) in which case \(\mathcal {C}_{n}\subseteq \mathcal {C}\). The last inclusion follows, for if \(G\in \mathcal {C}_{n} \), \(G\ne T \), then any non-trivial homomorphic image H of G must contain \(K_{n}\in \mathcal {C}\) as an induced subgraph. By definition of a connectedness, \(G\in \mathcal {C}\) follows. If the universal class \(\mathcal {W}\) contains at least one graph with an infinite vertex set, then \(\mathcal {C}:=\{G\in \mathcal {W}\mid G=T\) or \(E_{G}\) is infinite} and \(\mathcal {D}:=\{G\in \mathcal {W}\mid E_{G}\) is finite} is a corresponding pair of a connectedness and a disconnectedness respectively and \(\mathcal {C}_{n}\) is not contained in \(\mathcal {C}\) for all \(n\ge 1.\)