Isolation of Regular Graphs and k-Chromatic Graphs

Abstract

Given a set ${\mathcal {F}}$ of graphs, we call a copy of a graph in ${\mathcal {F}}$ an ${\mathcal {F}}$-graph. The ${\mathcal {F}}$-isolation number of a graph G, denoted by $\iota (G,{\mathcal {F}})$, is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the ${\mathcal {F}}$-graphs contained by G (equivalently, $G - N[D]$ contains no ${\mathcal {F}}$-graph). Thus, $\iota (G,\{K_1\})$ is the domination number of G. For any integer $k \ge 1$, let ${\mathcal {F}}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let ${\mathcal {F}}_{2,k}$ be the set of graphs whose chromatic number is at least k, and let ${\mathcal {F}}_{3,k}$ be the union of ${\mathcal {F}}_{1,k}$ and ${\mathcal {F}}_{2,k}$. Thus, k-cliques are members of both ${\mathcal {F}}_{1,k}$ and ${\mathcal {F}}_{2,k}$. We prove that for each $i \in \{1, 2, 3\}$, $\frac{m+1}{{k \atopwithdelims ()2} + 2}$ is a best possible upper bound on $\iota (G, {\mathcal {F}}_{i,k})$ for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.

1 Introduction

For standard terminology in graph theory, we refer the reader to [25]. Most of the terminology used in this paper is defined in [1].

The set of positive integers is denoted by ${\mathbb {N}}$. For $n \in \{0\} \cup {\mathbb {N}}$, [n] denotes the set $\{i \in {\mathbb {N}} :i \le n\}$. Note that [0] is the empty set $\emptyset$. Arbitrary sets and graphs are taken to be finite. For a non-empty set X, ${X \atopwithdelims ()2}$ denotes the set of 2-element subsets of X.

Every graph G is taken to be simple, meaning that its vertex set V(G) and edge set E(G) satisfy $E(G) \subseteq {V(G) \atopwithdelims ()2}$. An edge $\{v,w\}$ may be represented by vw. If $n = |V(G)|$, then G is called an n-vertex graph. If $m = |E(G)|$, then G is called an m-edge graph. For $v \in V(G)$, $N_{G}(v)$ denotes the set of neighbors of v in G, $N_{G}[v]$ denotes the closed neighborhood $N_{G}(v) \cup \{ v \}$ of v, and $d_{G}(v)$ denotes the degree $|N_{G} (v)|$ of v. For $X \subseteq V(G)$, $N_G[X]$ denotes the closed neighborhood $\bigcup _{v \in X} N_G[v]$ of X, G[X] denotes the subgraph of G induced by X, and $G-X$ denotes the graph obtained by deleting the vertices in X from G. Thus, $G[X] = (X,E(G) \cap {X \atopwithdelims ()2})$ and $G - X = G[V(G) \backslash X]$. With a slight abuse of notation, for $S \subseteq E(G)$, $G - S$ denotes the graph obtained by removing the edges in S from G, that is, $G - S = (V(G), E(G) \backslash S)$. For $x \in V(G) \cup E(G)$, $G - \{x\}$ may be abbreviated to $G - x$. The subscript G may be omitted where no confusion arises; for example, $N_G(v)$ may be abbreviated to N(v).

For $n \ge 1$, $K_n$ and $P_n$ denote the graphs $([n], {[n] \atopwithdelims ()2})$ and $([n], \{\{i,i+1\} :i \in [n-1]\})$, respectively. For $n \ge 3$, $C_n$ denotes $([n], \{\{1,2\}, \{2,3\}, \dots , \{n-1,n\}, \{n,1\}\})$. A copy of $K_n$ is called a complete graph or an n-clique. A copy of $P_n$ is called an n-path or simply a path. A copy of $C_n$ is called an n-cycle or simply a cycle. A 3-cycle is a 3-clique and is also called a triangle. If G contains a k-clique H, then we call H a k-clique of G.

If $D \subseteq V(G) = N[D]$, then D is called a dominating set of G. The domination number of G, denoted by $\gamma (G)$, is the size of a smallest dominating set of G. If ${\mathcal {F}}$ is a set of graphs and F is a copy of a graph in ${\mathcal {F}}$, then we call F an ${\mathcal {F}}$-graph. A subset D of V(G) is called an ${\mathcal {F}}$-isolating set of G if no subgraph of $G-N[D]$ is an ${\mathcal {F}}$-graph. The ${\mathcal {F}}$-isolation number of G, denoted by $\iota (G, {\mathcal {F}})$, is the size of a smallest ${\mathcal {F}}$-isolating set of G. We may abbreviate $\iota (G, \{F\})$ to $\iota (G, F)$. Clearly, D is a dominating set of G if and only if D is a $\{K_1\}$-isolating set of G. Thus, $\gamma (G) = \iota (G, K_1)$.

Caro and Hansberg introduced the isolation problem in [8]. It is a natural generalization of the classical domination problem [10, 11, 14,15,16,17]. Ore [22] proved that $\gamma (G) \le n/2$ for any connected n-vertex graph G with $n \ge 2$ (see [14]). This is one of the earliest results in domination theory. While the deletion of the closed neighborhood of a dominating set produces the null graph (the graph with no vertices), the deletion of the closed neighborhood of a $\{K_2\}$-isolating set produces an empty graph (a graph with no edges). Caro and Hansberg [8] proved that if G is a connected n-vertex graph, then $\iota (G, K_2) \le n/3$ unless G is a 2-clique or a 5-cycle. Solving a problem of Caro and Hansberg [8], Fenech, Kaemawichanurat, and the present author [2] proved that

$$\begin{aligned} \iota (G, K_k) \le \frac{n}{k+1} \end{aligned}$$

(1)

unless G is a k-clique or $k = 2$ and G is a 5-cycle, and that the bound is sharp. Ore’s result and the Caro–Hansberg result are the cases $k = 1$ and $k = 2$, respectively. Let ${\mathcal {C}}$ be the set of cycles. Solving another problem of Caro and Hansberg [8], the present author [1] proved that

$$\begin{aligned} \iota (G,{\mathcal {C}}) \le \frac{n}{4} \end{aligned}$$

(2)

unless G is a triangle, and that the bound is sharp. Domination and isolation have been particularly investigated for maximal outerplanar graphs [4, 5, 7,8,9, 12, 13, 18,19,20,21, 23, 24], mostly due to connections with Chvátal’s Art Gallery Theorem [9].

Fenech, Kaemawichanurat, and the present author [3] also obtained a sharp upper bound on $\iota (G,K_k)$ in terms of the number of edges. The result is the analog of (1) given by Theorem 1. We need the following construction from [3].

Construction 1

([3]). Consider any $m, k \in {\mathbb {N}}$. Let $t_{k} = {k \atopwithdelims ()2} + 2$ and $q = \lfloor \frac{m+1}{t_k} \rfloor$. Then, $m+1 = qt_k + r$ for some $r \in \{0\} \cup [t_k - 1]$. Let $Q_{m,k}$ be a q-element set. If $q \ge 1$, then let $v_1, \dots , v_q$ be the elements of $Q_{m,k}$, let $U_1, \dots , U_q$ be k-element sets such that $U_1, \dots , U_q, Q_{m,k}$ are pairwise disjoint, and for each $i \in [q]$, let $u_i^1, \dots , u_i^k$ be the elements of $U_i$, and let $G_i = (U_i \cup \{v_i\}, {U_i \atopwithdelims ()2} \cup \{u_i^1v_i\})$. If either $q = 0$, $T = (\emptyset , \emptyset )$, and G is an r-edge tree $T'$, or $q \ge 1$, T is a tree with $V(T) = Q_{m,k}$, $T'$ is an r-edge tree with $V(T') \cap \bigcup _{i=1}^q V(G_i) = \{v_q\}$, and $G = \left( V(T') \cup \bigcup _{i=1}^q V(G_i), E(T) \cup E(T') \cup \bigcup _{i=1}^q E(G_i) \right)$, then we say that G is an (m, k)-special graph with quotient graph T and remainder graph $T'$, and for each $i \in [q]$, we call $G_i$ and $v_i$ a k-clique constituent of G and the k-clique connection of $G_i$ in G, respectively. We say that an (m, k)-special graph is pure if $r = 0$. (Figure 1 in [3] is an illustration of a pure (m, k)-special graph.)

Theorem 1

([3]). If $k \ge 1$ and G is a connected m-edge graph that is not a k-clique, then

$$\begin{aligned} \iota (G,K_k) \le \frac{m+1}{{k\atopwithdelims ()2}+2}. \end{aligned}$$

(3)

Moreover,

• (i) Equality in (3) holds if and only if either G is a pure (m, k)-special graph or $k=2$ and G is a 5-cycle.

• (ii) If G is an (m, k)-special graph, then $\iota (G,K_k) = \left\lfloor (m+1)/\left( {k\atopwithdelims ()2}+2 \right) \right\rfloor .$

Problem 7.4 in [8] asks for bounds on $\iota (G,{\mathcal {F}})$ for other interesting sets ${\mathcal {F}}$. Section 1 of that seminal paper makes particular mention of k-colorable graphs. We address the problem for such graphs and the problem for regular graphs together. The main result presented here generalizes Theorem 1 in these two desirable directions.

If $V(G) \ne \emptyset$ and $d(v) = r$ for each $v \in V(G)$, then G is said to be r-regular or simply regular, and r is called the degree of G. For $k \ge 1$, let ${\mathcal {F}}_{1,k}$ be the set of regular graphs whose degree is at least $k-1$.

If there exists a function $f :V(G) \rightarrow [k]$ such that $f(v) \ne f(w)$ for every $v, w \in V(G)$ with $vw \in E(G)$, then G is said to be k-colorable, and f is called a proper k-coloring of G. The chromatic number of G, denoted by $\chi (G)$, is the smallest non-negative integer k such that G is k-colorable. If $k = \chi (G)$, then G is said to be k-chromatic. For $k \ge 1$, let ${\mathcal {F}}_{2,k}$ be the set of graphs whose chromatic number is at least k.

Let ${\mathcal {F}}_{3,k}$ be the union of ${\mathcal {F}}_{1,k}$ and ${\mathcal {F}}_{2,k}$. In Sect. 2, we prove the following result.

Theorem 2

If $\ell \in \{1, 2, 3\}$, $k \ge 1$, and G is a connected m-edge graph that is not a k-clique, then

$$\begin{aligned} \iota (G, {\mathcal {F}}_{\ell ,k}) \le \frac{m+1}{{k\atopwithdelims ()2}+2}. \end{aligned}$$

(4)

Moreover,

• (i) Equality in (4) holds if and only if G is a pure (m, k)-special graph, or $k=2$ and G is a 5-cycle, or $\ell \in \{1, 3\}$, $k = 3$, and G is a 4-cycle.

• (ii) If G is an (m, k)-special graph, then $\iota (G, {\mathcal {F}}_{\ell ,k}) = \left\lfloor (m+1)/\left( {k\atopwithdelims ()2}+2 \right) \right\rfloor .$

The proof of the bound depends on determining the graphs attaining the bound (i.e., the proof of (4) uses (i) and (ii)). This appears to be a new feature in the literature on isolation.

Since k-cliques are $(k-1)$-regular and cycles are 2-regular, Theorem 1 and the following analog of (2) are immediate consequences.

Theorem 3

If G is a connected m-edge graph that is not a triangle, then

$$\begin{aligned} \iota (G,{\mathcal {C}}) \le \frac{m+1}{5}. \end{aligned}$$

(5)

Moreover,

• (i) Equality in (5) holds if and only if G is a pure (m, 3)-special graph or a 4-cycle.

• (ii) If G is an (m, 3)-special graph, then $\iota (G,{\mathcal {C}}) = \left\lfloor (m+1)/5 \right\rfloor .$

2 Proof of Theorem 2

In this section, we prove Theorem 2. We start with two lemmas from [1].

Lemma 1

([1]). If G is a graph, ${\mathcal {F}}$ is a set of graphs, $X \subseteq V(G)$, and $Y \subseteq N[X]$, then

$$\begin{aligned} \iota (G, {\mathcal {F}}) \le |X| + \iota (G-Y, {\mathcal {F}}). \end{aligned}$$

The set of components of a graph G will be denoted by $\textrm{C}(G)$.

Lemma 2

([1]). If G is a graph and ${\mathcal {F}}$ is a set of graphs, then

$$\begin{aligned} \iota (G,{\mathcal {F}}) = \sum _{H \in \mathrm{C(G)}} \iota (H,{\mathcal {F}}). \end{aligned}$$

A basic result in graph theory is that for any connected graph G, $|E(G)| \ge |V(G)|-1$, and equality holds if and only if G is a tree.

Proposition 1

If G is a pure (m, k)-special graph with exactly q k-clique constituents, then $m = ({k\atopwithdelims ()2}+2)q-1$ and $\iota (G,{\mathcal {F}}_{\ell ,k}) = q$ for each $\ell \in \{1, 2, 3\}$.

Proof

Suppose that G is a pure (m, k)-special graph with exactly q k-clique constituents as in Construction 1. Then, $\{v_1, \dots , v_q\}$ is an ${\mathcal {F}}_{\ell ,k}$-isolating set of G, so $\iota (G,{\mathcal {F}}_{\ell ,k}) \le q$. The subgraphs $G[U_1], \dots , G[U_q]$ of G are k-cliques, which are $(k-1)$-regular and k-chromatic. Thus, if D is an ${\mathcal {F}}_{\ell ,k}$-isolating set of G, then $D \cap V(G_i) \ne \emptyset$ for each $i \in [q]$. Therefore, $\iota (G,{\mathcal {F}}_{\ell ,k}) = q$. Now, clearly $m = ({k\atopwithdelims ()2}+1)q + |E(T)|$. Since T is a q-vertex tree, $|E(T)| = q-1$. Thus, $m = ({k\atopwithdelims ()2}+2)q-1$. $\square$

If G is a copy of a graph H, then we write $G \simeq H$. For a graph G, the maximum degree of G (that is, $\max \{d(v) :v \in V(G)\}$) is denoted by $\Delta (G)$.

We will use the following classical result.

Theorem 4

(Brooks’ Theorem [6]). If G is a connected n-vertex graph, then $\chi (G) \le \Delta (G)$ unless $G \simeq K_n$ or n is odd and $G \simeq C_n$.

Proof of Theorem 2

Let $t_k = {k \atopwithdelims ()2} + 2$ and ${\mathcal {F}} = {\mathcal {F}}_{3,k}$.

The argument in the proof of Proposition 1 yields (ii). If G is a 4-cycle, then $\iota (G,{\mathcal {F}}_{\ell ,1}) = 2 < \frac{5}{2} = \frac{m+1}{t_1}$, $\iota (G,{\mathcal {F}}_{\ell ,2}) = 1 < \frac{m+1}{t_2}$, $\iota (G,{\mathcal {F}}_{1,3}) = \iota (G,{\mathcal {F}}_{3,3}) = 1 = \frac{m+1}{t_3}$, $\iota (G,{\mathcal {F}}_{2,3}) = 0 < \frac{m+1}{t_3}$, and if $k \ge 4$, then $\iota (G,{\mathcal {F}}_{\ell ,k}) = 0 < \frac{m+1}{t_k}$. If G is a 5-cycle, then $\iota (G,{\mathcal {F}}_{\ell ,1}) = 2 < \frac{6}{2} = \frac{m+1}{t_1}$, $\iota (G,{\mathcal {F}}_{\ell ,2}) = 2 = \frac{m+1}{t_2}$, $\iota (G,{\mathcal {F}}_{\ell ,3}) = 1 < \frac{m+1}{t_3}$, and if $k \ge 4$, then $\iota (G,{\mathcal {F}}_{\ell ,k}) = 0 < \frac{m+1}{t_k}$. Together with Proposition 1, this settles the sufficiency condition in (i).

Using induction on m, we now prove that the bound in (4) holds, and we prove that it is attained only if G and k are as in (i). For the bound, since $\iota (G, {\mathcal {F}})$ is an integer and $\iota (G,{\mathcal {F}}_{\ell ,k}) \le \iota (G,{\mathcal {F}})$, it suffices to prove that $\iota (G, {\mathcal {F}}) \le \frac{m+1}{t_k}$. If $k \le 2$, then a $\{K_k\}$-isolating set of G is an ${\mathcal {F}}_{\ell ,k}$-isolating set of G, so the result is given by Theorem 1. Consider $k \ge 3$. The result is trivial if $m \le 2$ or $\iota (G,{\mathcal {F}}) = 0$. Suppose that $m \ge 3$, $\iota (G,{\mathcal {F}}) \ge 1$, and G is not a k-clique. Let $n = |V(G)|$. Since G is connected, $m \ge n - 1$, so $n \le m + 1$.

Case 1: G has no k-cliques.

Subcase 1.1: $d(v) \le k-2$ for some $v \in V(G)$. Let $G' = G-v$. Let $H_1, \dots , H_r$ be the distinct components of $G'$.

Consider any $i \in [r]$. Let $D_i$ be a smallest ${\mathcal {F}}$-isolating set of $H_i$. Let $H_i' = H_i - N[D_i]$, let $c_i = \chi (H_i')$, and let $f_i$ be a proper $c_i$-coloring of $H_i'$. We have that $c_i \le k-1$ and that $H_i'$ contains no regular graph of degree at least $k-1$. Since G has no k-cliques, $|D_i| \le \frac{|E(H_i)|+1}{t_k}$ by the induction hypothesis.

Let $D = \bigcup _{i=1}^r D_i$. Suppose that $G - N[D]$ contains a connected regular graph R of degree at least $k-1$. Since $d_R(v) \le d_G(v)\le k-2$, $v \notin R$. We obtain that for some $i \in [r]$, $H_i'$ contains R, a contradiction. Thus, $G - N[D]$ contains no regular graph of degree at least $k-1$. Let $f' :V(G'-N_{G'}[D]) \rightarrow [k-1]$ such that $f'(w) = f_i(w)$ for each $i \in [r]$ and each $w \in V(H_i')$. Since $d(v) \le k-2$, $[k-1]$ has an element j that is not in $\{f'(w) :w \in N(v) \cap V(G' - N_{G'}[D])\}$. Let $f :V(G-N[D]) \rightarrow [k-1]$ such that $f(w) = f'(w)$ for each $w \in V(G'-N_{G'}[D])$, and such that if $v \in V(G-N[D])$, then $f(v) = j$. Since f is a proper $(k-1)$-coloring of $G - N[D]$, D is an ${\mathcal {F}}$-isolating set of G.

Since G is connected, for each $i \in [r]$, $vw_i \in E(G)$ for some $w_i \in V(H_i)$. We have

$$\begin{aligned} m \ge \left| \bigcup _{i=1}^r (\{vw_i\} \cup E(H_i))\right| \ge \sum _{i=1}^r t_k|D_i| = t_k|D| \ge t_k \iota (G,{\mathcal {F}}), \end{aligned}$$

so $\iota (G,{\mathcal {F}}) < \frac{m+1}{t_k}$.

Subcase 1.2: $d(v) \ge k-1$ for each $v \in V(G)$. Then, $\Delta (G) \ge k-1$. Let $v \in V(G)$ with $d(v) = \Delta (G)$. Let $G' = G - N[v]$.

Suppose $d(v) = k-1$. Then, G is $(k-1)$-regular. Since G is not a k-clique, we have $n \ge k+1$, so $V(G') \ne \emptyset$. Suppose that $G'$ contains a $(k-1)$-regular graph $R'$. Since G is connected, $uw \in E(G)$ for some $u \in V(R')$ and some $w \in V(G) \backslash V(R')$. We have $d_G(u) \ge d_{R'}(u) + 1 \ge k$, which contradicts $\Delta (G) = k-1$. Thus, $G'$ contains no $(k-1)$-regular graph. Since $\Delta (G') \le \Delta (G) = k-1$, $\chi (G') \le k-1$ by Theorem 4. Thus, $\{v\}$ is an ${\mathcal {F}}$-isolating set of G. By the handshaking lemma, we have $2\,m = (k-1)n \ge (k-1)(k+1) = 2t_k + k - 5$, so $m + 1 \ge t_k + \frac{k-3}{2} \ge t_k$. We have $\iota (G, {\mathcal {F}}) = 1 \le \frac{m+1}{t_k}$. Suppose $\iota (G, {\mathcal {F}}) = \frac{m+1}{t_k}$. Then, $n = k+1$ and $k = 3$. Thus, G is a 4-cycle.

Now suppose $d(v) \ge k$. Let $s = d(v)$. Suppose $N[v] = V(G)$. By the handshaking lemma,

$$\begin{aligned} 2m&= d(v) + \sum _{x \in N(v)} d(x) \ge d(v) + d(v)(k-1) \ge k + k(k-1) = 2t_k + k-4 \\&\ge 2t_k - 1, \end{aligned}$$

so $t_k < m+1$. Since $\{v\}$ is an ${\mathcal {F}}$-isolating set of G, $\iota (G,{\mathcal {F}}) = 1 < \frac{m+1}{t_k}$. Now suppose $N[v] \ne V(G)$. Then, $V(G') \ne \emptyset$. Let $H_0 = G[N(v)]$, and let $H_1, \dots , H_r$ be the distinct components of $G'$. Since G is connected, for each $i \in [r]$, $x_iy_i \in E(G)$ for some $x_i \in N(v)$ and $y_i \in V(H_i)$. Let $A = \{xy \in E(G) :x \in N(v), \, y \in V(G')\}$. Then, $x_1y_1, \dots , x_ry_r \in A$, so $|A| \ge r$. By the handshaking lemma,

$$\begin{aligned} 2|E(H_0)|&= \sum _{x \in N(v)} d_{H_0}(x) = \sum _{x \in N(v)} |N(x) \backslash (\{v\} \cup V(G'))| \\&= \sum _{x \in N(v)} (d(x) - 1 - |N(x) \cap V(G')|) \\&\ge (k-2)d(v) - \sum _{x \in N(v)} |N(x) \cap V(G')| = (k-2)s - |A|, \end{aligned}$$

so $|E(H_0)| \ge \frac{(k-2)s - |A|}{2}$. Let $A' = \{vx :x \in N(v)\} \cup E(H_0) \cup A$. Then,

$$\begin{aligned} |A'|&= s + |E(H_0)| + |A| \ge \frac{sk + |A|}{2} = \frac{(k-1)k}{2} + \frac{(s-k+1)k + |A|}{2} \\&= t_k + \frac{(s-k+1)k + |A| - 4}{2}. \end{aligned}$$

Since $E(G) = A' \cup \bigcup _{i=1}^r E(H_i)$ and $|A| \ge r$,

$$\begin{aligned} m = |A'| + \sum _{i=1}^r |E(H_i)| \ge t_k + \frac{(s-k+1)k + r - 4}{2} + \sum _{i=1}^r |E(H_i)|. \end{aligned}$$

(6)

For each $i \in [r]$, let $D_i$ be a smallest ${\mathcal {F}}$-isolating set of $H_i$. Let $D = \{v\} \cup \bigcup _{i=1}^r D_i$. Then, D is an ${\mathcal {F}}$-isolating set of G. Since G has no k-cliques, for each $i \in [r]$, $H_i$ is not a pure $(|E(H_i)|,k)$-special graph.

Suppose $k \ne 3$ or $H_i \not \simeq C_4$ for each $i \in [r]$. By the induction hypothesis, for each $i \in [r]$, $|D_i| < \frac{|E(H_i)|+1}{t_k}$, and hence, since $t_k|D_i| < |E(H_i)| + 1$, $t_k|D_i| \le |E(H_i)|$. By (6),

$$\begin{aligned} m \ge t_k \left( 1 + \sum _{i=1}^r |D_i| \right) + \frac{(s-k+1)k + r - 4}{2} \ge t_k|D| + \frac{k + r - 4}{2} \ge t_k |D| \end{aligned}$$

as $k \ge 3$ and $r \ge 1$. We have $\iota (G, {\mathcal {F}}) \le |D| < \frac{m+1}{t_k}$.

Now suppose $k = 3$ and $H_j \simeq C_4$ for some $j \in [r]$. We may assume that $j = 1$. Let $z_1$ and $z_2$ be the two members of $N_{H_1}(y_1)$, and let $z_3$ be the member of $V(H_1) \backslash N_{H_1}[y_1]$. We have $V(H_1) = \{y_1, z_1, z_2, z_3\}$. Let $G_1 = G - \{y_1, z_1, z_2\}$. Note that $N_{G_1}(v) = N_G(v)$. If $d_{G_1}(z_3) = 0$, then the components of $G_1$ are $(\{z_3\}, \emptyset )$ and $G_1 - z_3$, and we take I to be $G_1 - z_3$. If $d_{G_1}(z_3) \ge 1$, then $G_1$ is connected, and we take I to be $G_1$. Since $d_{I}(v) = d_G(v) \ge k = 3$, I is not a 4-cycle. Since G has no k-cliques, I is not a pure (|E(I)|, k)-special graph. By the induction hypothesis, $\iota (I, {\mathcal {F}}) < \frac{|E(I)|+1}{t_k} = \frac{|E(I)|+1}{5}$. Let $D_I$ be an ${\mathcal {F}}$-isolating set of I of size $\iota (I, {\mathcal {F}})$. Since $\{y_1, z_1, z_2\} \subset N[y_1]$, $\{y_1\} \cup D_I$ is an ${\mathcal {F}}$-isolating set of G. Let $J = \{x_1y_1, y_1z_1, y_1z_2, z_1z_3, z_2z_3\}$. Note that $J \subseteq E(G) \backslash E(I)$. We have

$$\begin{aligned} \iota (G,{\mathcal {F}}) \le 1 + |D_I| < 1 + \frac{|E(I)|+1}{5} = \frac{|E(I)| + |J| + 1}{5} \le \frac{m+1}{5} = \frac{m+1}{t_k}. \end{aligned}$$

Case 2: G has at least one k-clique. Let F be a k-clique of G. Since G is connected and $G \not \simeq K_k$, $N[v] \backslash V(F) \ne \emptyset$ for some $v \in V(F)$. Let $u \in N[v] \backslash V(F)$. We have $V(F) \subseteq N[v]$, so $|N[v]| \ge k+1$ and

$$\begin{aligned} |E(G[N[v]])| \ge t_k - 1. \end{aligned}$$

(7)

Suppose $V(G) = N[v]$. Then, $\{v\}$ is an ${\mathcal {F}}$-isolating set of G, so $\iota (G,{\mathcal {F}}) = 1 \le \frac{m+1}{t_k}$ by (7). If $\iota (G,{\mathcal {F}}) = \frac{m+1}{t_k}$, then $V(G) = V(F) \cup \{u\}$ and $E(G) = E(F) \cup \{uv\}$, so G is a pure (m, k)-special graph.

Now suppose $V(G) \ne N[v]$. Let $G' = G-N[v]$ and $n' = |V(G')|$. We have

$$\begin{aligned} n \ge n' + k + 1 \end{aligned}$$

(8)

and $n' \ge 1$. Let ${\mathcal {H}} = \textrm{C}(G')$. For any $H \in {\mathcal {H}}$ and $x \in N(v)$ such that $xy \in E(G)$ for some $y \in V(H)$, we say that H is linked to x. Since G is connected, for each $H \in {\mathcal {H}}$, H is linked to at least one member of N(v), so $x_Hy_H \in E(G)$ for some $x_H \in N(v)$ and some $y_H \in V(H)$. Clearly,

$$\begin{aligned} m \ge |E(F) \cup \{ uv \}| + \sum _{H \in {\mathcal {H}}} |E(H) \cup \{x_Hy_H\}| = t_k - 1 + \sum _{H \in {\mathcal {H}}} (|E(H)| + 1).\nonumber \\ \end{aligned}$$

(9)

Let

$$\begin{aligned} {\mathcal {H}}' = \{H \in {\mathcal {H}} :H \simeq K_k\}. \end{aligned}$$

By the induction hypothesis, for each $H \in {\mathcal {H}} \backslash {\mathcal {H}}'$, $\iota (H,{\mathcal {F}}) \le \frac{|E(H)| + 1}{t_k}$, and equality holds only if H is a pure (|E(H)|, k)-special graph or $k = 3$ and H is a 4-cycle.

Subcase 2.1: ${\mathcal {H}}' = \emptyset$. By Lemma 1 (with $X = \{v\}$ and $Y = N[v]$), Lemma 2, and (9),

$$\begin{aligned} \iota (G,{\mathcal {F}})&\le 1 + \iota (G',{\mathcal {F}}) = 1 + \sum _{H \in {\mathcal {H}}} \iota (H,{\mathcal {F}}) \le 1 + \sum _{H \in {\mathcal {H}}} \frac{|E(H)|+1}{t_k} \le \frac{m+1}{t_k}.\nonumber \\ \end{aligned}$$

(10)

Suppose $\iota (G,{\mathcal {F}}) = \frac{m+1}{t_k}$. Then, the inequality in (9) is an equality, and $\iota (H,{\mathcal {F}}) = \frac{|E(H)| + 1}{t_k}$ for each $H \in {\mathcal {H}}$.

Let $H_1, \dots , H_r$ be the distinct members of ${\mathcal {H}}$. Let $I = \{i \in [r] :H_i$ is a pure $(|E(H_i)|,k)$-special graph$\}$. For each $i \in I$, let $G_{i,1}, \dots , G_{i,q_i}$ be the distinct k-clique constituents of $H_i$, and for each $j \in [q_i]$, let $v_{i,j}$ be the k-clique connection of $G_{i,j}$ in $H_i$, and let $u_{i,j}^1, \dots , u_{i,j}^k$ be the vertices of $G_{i,j} - v_{i,j}$, where $u_{i,j}^1$ is the neighbor of $v_{i,j}$ in $G_{i,j}$. By Proposition 1, $|E(H_i)| = t_kq_i - 1$ for each $i \in I$. If $i \in [r] \backslash I$, then, by the above, $k = 3$, $H_i$ is a 4-cycle, and $|E(H_i)| + 1 = t_k$. Since equality holds throughout in (9), we have

$$\begin{aligned} V(G) = V(F) \cup \{u\} \cup \bigcup _{i=1}^r V(H_i) \end{aligned}$$

(11)

and

$$\begin{aligned} E(G) = E(F) \cup \{uv\} \cup \{x_{H_1}y_{H_1}, \dots , x_{H_r}y_{H_r}\} \cup \bigcup _{i=1}^r E(H_i). \end{aligned}$$

(12)

Let

$$\begin{aligned} D = \{v\} \cup \{y_{H_i} :i \in [r] \backslash I\} \cup \left( \bigcup _{i \in I} \bigcup _{j=1}^{q_i} \{u_{i,j}^1\} \right) . \end{aligned}$$

Then, D is an ${\mathcal {F}}$-isolating set of G, and

$$\begin{aligned} |D|&= 1 + \left( \sum _{i \in [r] \backslash I} \frac{|E(H_i)| + 1}{t_k} \right) + \sum _{i \in I} q_i = 1 + \sum _{i=1}^{r} \frac{|E(H_i)|+1}{t_k} = \frac{m+1}{t_k}\\&= \iota (G,{\mathcal {F}}). \end{aligned}$$

Suppose that $[r] \backslash I$ has an element j. Then, $k = 3$ and $H_j$ is a 4-cycle. Suppose $x_{H_j} = u$. Then, we have that $(D \backslash \{v, y_{H_j}\}) \cup \{u\}$ is an ${\mathcal {F}}$-isolating set of G of size $|D|-1$, contradicting $|D| = \iota (G,{\mathcal {F}})$. Thus, $x_{H_j} \in V(F)$. This yields that $D \backslash \{v\}$ is an ${\mathcal {F}}$-isolating set of G, contradicting $|D| = \iota (G,{\mathcal {F}})$.

Therefore, $[r] \backslash I = \emptyset$, and hence $I = [r]$. Consider any $i \in [r]$. Suppose $x_{H_i} \in V(F)$ and $y_{H_i} = v_{i,j}$ for some $j \in [q_i]$. Then, the components of $G - N[(D \backslash \{v, u_{i,j}^1\}) \cup \{v_{i,j}\}]$ are $G_{i,j} - \{v_{i,j}, u_{i,j}^1\}$ and $G[\{u\} \cup V(F-x_{H_i})]$, and clearly they contain no $\mathcal {F}$-graphs (a proper $(k-1)$-coloring of $G[\{u\} \cup V(F-x_{H_i})]$ is obtained by assigning different colors to the $k-1$ vertices of $F - x_{H_i}$, and coloring u differently from v). Thus, we have that $(D \backslash \{v, u_{i,j}^1\}) \cup \{v_{i,j}\}$ is an ${\mathcal {F}}$-isolating set of G of size $|D| - 1$, a contradiction. Suppose $x_{H_i} \in V(F)$ and $y_{H_i} = u_{i,j}^l$ for some $j \in [q_i]$ and $l \in [k]$. As above, we obtain that $(D \backslash \{v, u_{i,j}^1\}) \cup \{u_{i,j}^l\}$ is an ${\mathcal {F}}$-isolating set of G of size $|D| - 1$, a contradiction. Thus, $x_{H_i} = u$. Suppose $y_{H_i}= u_{i,j}^l$ for some $j \in [q_i]$ and $l \in [k]$. Then, we have that $(D \backslash \{v,u_{i,j}^1\}) \cup \{u\}$ is an ${\mathcal {F}}$-isolating set of G of size $|D| - 1$, a contradiction. Thus, $y_{H_i}= v_{i,j}$ for some $j \in [q_i]$. Since this has been established for each $i \in [r]$, we have that $G[\{u\} \cup \bigcup _{i=1}^r \bigcup _{j=1}^{q_i} \{v_{i,j}\}]$ is a tree, and hence G is a pure (m, k)-special graph with $G[N[v]], G_{1,1}, \dots , G_{1,q_1}, \dots , G_{r,1}, \dots , G_{r,q_r}$ being its k-clique constituents and u being the k-clique connection of G[N[v]] in G.

Subcase 2.2: ${\mathcal {H}}' \ne \emptyset$. Let $H' \in {\mathcal {H}}'$. Then, $H'$ is linked to some $x \in N(v)$. Let

${\mathcal {H}}_{1} = \{H \in {\mathcal {H}}' :H$ is only linked to $x\}$ and

${\mathcal {H}}_{2} = \{H \in {\mathcal {H}} \backslash {\mathcal {H}}' :H$ is only linked to $x\}$.

For each $H \in {\mathcal {H}}_2$, let $D_H$ be a smallest ${\mathcal {F}}$-isolating set of H. Then, $|D_H| = \iota (H,{\mathcal {F}}) \le \frac{|E(H)|+1}{t_k}$ by the induction hypothesis.

Subcase 2.2.1: Each member of ${\mathcal {H}}'$ is linked to at least two members of N(v). Thus, ${\mathcal {H}}_1 = \emptyset$. Let

$$\begin{aligned} X = \{x\} \cup V(H') \quad \text {and} \quad G^* = G - X. \end{aligned}$$

Then, a component $G^*_v$ of $G^*$ satisfies $N[v] \backslash \{x\} \subseteq V(G^*_v)$, and the members of ${\mathcal {H}}_2$ are the other components of $G^*$. Let $D^*_v$ be a smallest ${\mathcal {F}}$-isolating set of $G^*_v$. Since $H'$ is linked to x and to some $x' \in N(v) \backslash \{x\}$, there exist $y, y' \in V(H')$ with $xy, x'y' \in E(G)$. Let $D = D^*_v \cup \{y\} \cup \bigcup _{H \in {\mathcal {H}}_2} D_H$. Since $X \subseteq N[y]$, D is an ${\mathcal {F}}$-isolating set of G. Thus,

$$\begin{aligned} \iota (G,{\mathcal {F}}) \le |D^*_v| + 1 + \sum _{H \in {\mathcal {H}}_2} |D_H| \le |D^*_v| + \frac{t_k}{t_k} + \sum _{H \in {\mathcal {H}}_2} \frac{|E(H)| + 1}{t_k}. \end{aligned}$$

We have

$$\begin{aligned} m\ge & {} |E(G^*_v) \cup \{vx\}| + |E(H') \cup \{xy, x'y'\}| + \sum _{H \in {\mathcal {H}}_2} |E(H) \cup \{x_Hy_H\}| \nonumber \\= & {} |E(G^*_v)| + 1 + t_k + \sum _{H \in {\mathcal {H}}_2} (|E(H)| +1). \end{aligned}$$

(13)

If $G^*_v \not \simeq K_k$, then $|D^*_v| \le \frac{|E(G^*_v)|+1}{t_k}$ by the induction hypothesis, so

$$\begin{aligned} \iota (G,{\mathcal {F}}) \le \frac{|E(G^*_v)| + 1}{t_k} + \frac{t_k}{t_k} + \sum _{H \in {\mathcal {H}}_2} \frac{|E(H)| + 1}{t_k} < \frac{m+1}{t_k}. \end{aligned}$$

Now suppose $G^*_v \simeq K_k$. Since $|N[v] \backslash \{x\}| \ge k$ and $N[v] \backslash \{x\} \subseteq V(G^*_v)$, $V(G^*_v) = N[v] \backslash \{x\}$. Let $Y = (X \cup V(G^*_v)) \backslash \{v,x,y\}$ and $G_Y = G - \{v,x,y\}$. Then, the components of G[Y] and the members of ${\mathcal {H}}_2$ are the components of $G_Y$.

Suppose that G[Y] contains no ${\mathcal {F}}$-graph. Since $v, y \in N[x]$, $\{x\} \cup \bigcup _{H \in {\mathcal {H}}_2} D_H$ is an ${\mathcal {F}}$-isolating set of G, so $\iota (G,{\mathcal {F}}) \le 1 + \sum _{H \in {\mathcal {H}}_2} |D_H|$. Since $G^*_v \simeq K_k$, (13) gives us $m + 1 \ge 2t_k + \sum _{H \in {\mathcal {H}}_2} (|E(H)|+1) > t_k(1 + \sum _{H \in {\mathcal {H}}_2} |D_H|) \ge t_k\iota (G,{\mathcal {F}})$, so $\iota (G,{\mathcal {F}}) < \frac{m+1}{t_k}$.

Now suppose that G[Y] contains an ${\mathcal {F}}$-graph $F_Y$. Then, $\chi (F_Y) \ge k$ or $F_Y$ is a regular graph of degree at least $k-1$. Thus, $|N_{G[Y]}[z]| \ge k$ for some $z \in V(F_Y)$; this is given by Theorem 4 if $\chi (F_Y) \ge k$. Let $W \subseteq N_{G[Y]}[z]$ such that $z \in W$ and $|W| = k$. Let $G_1 = G_v^*$, $G_2 = H'$, $v_1 = v$, $v_2 = y$, $G_1' = G_1 - v_1$, and $G_2' = G_2 - v_2$. We have

$$\begin{aligned} N_{G[Y]}[z] \subseteq Y = V(G_1') \cup V(G_2'). \end{aligned}$$

(14)

Since $|V(G_1')| = |V(G_2')| = k-1$, it follows that $|W \cap V(G_1')| \ge 1$ and $|W \cap V(G_2')| \ge 1$. By (14), $z \in V(G_j')$ for some $j \in \{1, 2\}$. Let $Z = V(G_j) \cup W$. Since $z \in V(G_j)$ and $G_j \simeq K_k$,

$$\begin{aligned} Z \subseteq N[z]. \end{aligned}$$

(15)

We have

$$\begin{aligned} |Z| = |V(G_j)| + |W \backslash V(G_j)| = k + |W \cap V(G_{3-j}')| \ge k+1. \end{aligned}$$

(16)

Let $G_Z = G - Z$. Then, $V(G_Z) = \{x\} \cup (V(G_{3-j}) \backslash W) \cup \bigcup _{H \in {\mathcal {H}}_2} V(H)$. The components of $G_Z - x$ are the clique $G_Z[V(G_{3-j}) \backslash W]$ (which has less than k vertices) and the members of ${\mathcal {H}}_2$. Moreover, $v_{3-j} \in V(G_{3-j}) \backslash W$ (by (14)), $v_{3-j} \in N_{G_Z}(x)$, and $N_{G_Z}(x) \cap V(H) \ne \emptyset$ for each $H \in {\mathcal {H}}_2$ (by the definition of ${\mathcal {H}}_2$). Thus, $G_Z$ is connected.

Suppose ${\mathcal {H}}_2 \ne \emptyset$. Then, $G_Z \not \simeq K_k$. By (15), Lemma 1, and the induction hypothesis, $\iota (G,{\mathcal {F}}) \le 1 + \iota (G_Z,{\mathcal {F}}) \le 1 + \frac{|E(G_Z)|+1}{t_k}$. Let $w^* \in W \cap V(G_{3-j}')$ (recall that $|W \cap V(G_{3-j}')| \ge 1$). Since $G_{3-j} \simeq K_k$, $w^*w \in E(G_{3-j})$ for each $w \in V(G_{3-j}) \backslash \{w^*\}$. We have

$$\begin{aligned} m&\ge |E(G_j)| + |\{xv_j, zw^*\}| + |\{w^*w :w \in V(G_{3-j}) \backslash \{w^*\}\}| + |E(G_Z)| \\&\ge {k \atopwithdelims ()2} + 1 + k + |E(G_Z)| \ge t_k + 2 + |E(G_Z)| \end{aligned}$$

as $k \ge 3$. Thus, since $\iota (G,{\mathcal {F}}) \le 1 + \frac{|E(G_Z)|+1}{t_k}$, $\iota (G,{\mathcal {F}}) < \frac{m+1}{t_k}$.

Now suppose ${\mathcal {H}}_2 = \emptyset$. Then, $G^* = G^*_v$, so $V(G) = V(G^*_v) \cup \{x\} \cup V(H')$. Recall that the k-clique $H'$ is linked to at least two members of N(v). Thus,

$$\begin{aligned} n = 2k+1 \quad \text{ and } \quad m \ge 2{k \atopwithdelims ()2} + 3 = 2t_k - 1. \end{aligned}$$

If $|N[w]| \ge k+2$ for some $w \in V(G)$, then $|V(G-N[w])| \le k-1$, so $\{w\}$ is an ${\mathcal {F}}$-isolating set of G, and hence $\iota (G,{\mathcal {F}}) = 1 < \frac{m+1}{t_k}$. Suppose $|N[w]| \le k+1$ for each $w \in V(G)$. Then, $\Delta (G) = d(v) = k$, $N[z] = Z = V(G_j) \cup \{w\}$ for some $w \in V(G_{3-j}')$ (by (15) and (16)), and $V(G-N[z]) = \{x\} \cup V(G_{3-j} - w)$. If $G-N[z]$ contains no ${\mathcal {F}}$-graph, then $\iota (G,{\mathcal {F}}) = 1 < \frac{m+1}{t_k}$. Suppose that $G-N[z]$ contains an ${\mathcal {F}}$-graph $F'$. Since $|V(G-N[z])| = k$, $G-N[z] = F' \simeq K_k$. Since $\Delta (G) = k$, we have $N(x) = \{v_{j}\} \cup V(G_{3-j} - w)$ and, since $z \in V(G_j)$ and $w \in N[z] \cap V(G_{3-j})$, $N[w] = \{z\} \cup V(G_{3-j})$. Thus, $V(G-N[w]) = \{x\} \cup (V(G_j) \backslash \{z\})$. Since $|V(G-N[w])| = k \ge 3$ and $N[x] \cap V(G_j') = \emptyset$, $\{w\}$ is an ${\mathcal {F}}$-isolating set of G, so $\iota (G,{\mathcal {F}}) = 1 < \frac{m+1}{t_k}$.

Subcase 2.2.2: Some member of ${\mathcal {H}}'$ is linked to only one member of N(v). Thus, we may assume that $H'$ is linked to x only. For each $H \in {\mathcal {H}}_1 \cup {\mathcal {H}}_2$, $y_H \in N(x)$ for some $y_H \in V(H)$. By the same argument in the corresponding Subcase 2.2.2 of the proof of Theorem 1.3 in [3], we obtain that $\iota (G,{\mathcal {F}}) \le \frac{m+1}{t_k}$, and that equality holds only if

$$\begin{aligned} E(G) = E(H') \cup \{xy_{H'}\} \cup \bigcup _{H \in {\mathcal {H}}_2'} (E(H) \cup \{xy_H\}), \end{aligned}$$

where ${\mathcal {H}}_2' = {\mathcal {H}}_2 \cup \{G_v^*\}$ for a connected subgraph $G_v^*$ of G with

$$\begin{aligned} y_{G_v^*} = v \in V(G_v^*) = V(G) \backslash \left( \{x\} \cup V(H') \cup \bigcup _{H \in {\mathcal {H}}_2} V(H) \right) , \end{aligned}$$

and $\iota (H,{\mathcal {F}}) = \frac{|E(H)| + 1}{t_k}$ for each $H \in {\mathcal {H}}_2'$. Thus, if $\iota (G,{\mathcal {F}}) = \frac{m+1}{t_k}$, then G is as in the case $\iota (G,{\mathcal {F}}) = \frac{m+1}{t_k}$ in Subcase 2.1 (with $H'$ and x now taking the role of F and u, respectively, in (11) and (12)), so G is a pure (m, k)-special graph. $\square$