Lit-only sigma-game on nondegenerate graphs

A configuration of the lit-only $\sigma$-game on a graph $\Gamma$ is an assignment of one of two states, {\it on} or {\it off}, to each vertex of $\Gamma.$ Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allows the player to choose an {\it on} vertex $s$ of $\Gamma$ and change the states of all neighbors of $s.$ Given an integer $k$, the underlying graph $\Gamma$ is said to be $k$-lit if for any configuration, the number of {\it on} vertices can be reduced to at most $k$ by a finite sequence of moves. We give a description of the orbits of the lit-only $\sigma$-game on nondegenerate graphs $\Gamma$ which are not line graphs. We show that these graphs $\Gamma$ are 2-lit and provide a linear algebraic criterion for $\Gamma$ to be 1-lit.


Introduction
The notion of the σ-game on finite directed graphs Γ without multiple edges was first introduced by Sutner [16] in 1989. A configuration of the σ-game on Γ is an assignment of one of two states, on or off, to each vertex of Γ. Given a configuration, a move consists of choosing a vertex of Γ, followed by changing the states of all of its neighbors. If only on vertex is allowed to choose in each move, we come to the variation: lit-only σ-game. Starting from an initial configuration, the goal of the lit-only σ-game on Γ is to minimize the number of on vertices of Γ, or to reach an assigned configuration by a finite sequence of moves.
Given an integer k, the underlying graph Γ is said to be k-lit if for any configuration, the number of on vertices can be reduced to at most k by a finite sequence of the moves. More precisely, we are interested in the orbits of the lit-only σ-game on Γ and the smallest integer k, the minimum light number of Γ [18], for which Γ is k-lit. The notion of lit-only σ-games occurred implicitly in the study of equivalence classes of Vogan diagrams. The Borel-de Siebenthal theorem [2] showed that every Vogan diagram is equivalent to one with a single painted vertex, which implies that each simply-laced Dynkin diagram is 1-lit. The equivalence classes of Vogan diagrams were described by Chuah and Hu [7]. A conjecture made by Chang [5,6] that any tree with k leaves is ⌈k/2⌉-lit was confirmed by Wang and Wu [18], where the name "lit-only σ-game" was coined.
The lit-only σ-game on a simple graph Γ is simply the natural action of a certain subgroup H Γ of the general linear group over F 2 [18]. Under the assumption that Γ is the line graph of a simple graph G, Wu [19] described the orbits of the lit-only σ-game on Γ and gave a characterization for the minimum light number of Γ. Moreover, if G is a tree of order n ≥ 3, Wu showed that H Γ is isomorphic to the symmetric group on n letters. Weng and the author [12] determined the structure of H Γ without any assumption on G. The lit-only σ-game on a simple graph Γ can also be considered as a representation κ Γ of the simply-laced Coxeter group W Γ over F 2 [11]. The dual representation of κ Γ preserves a certain symplectic form B Γ . The two representations are equivalent whenever the form B Γ is nondegenerate. From this viewpoint it is natural to partition simple connected graphs into two classes according as B Γ is degenerate or nondegenerate.
In this paper we treat nondegenerate graphs Γ which are not line graphs. We show that H Γ is isomorphic to an orthogonal group, followed by a description of the orbits of lit-only σ-game on Γ (Theorem 3.1). Moreover we show that these graphs Γ are 2-lit and provide a linear algebraic criterion for Γ to be 1-lit (Theorem 3.2). Combining Theorem 3.1, Theorem 3.2 and those in [12] and [19], the study of the lit-only σ-game on nondegenerate graphs is quite completed and the focus for further research is on degenerate graphs.

Preliminaries
From now on, let Γ = (S, R) denote a finite simple connected graph with vertex set S and edge set R. Let F 2 denote the two-element field {0, 1}. Let V denote a F 2 -vector space that has a basis {α s | s ∈ S} in one-to-one correspondence with S.
for all t ∈ S. The set {f s | s ∈ S} forms a basis of V * and is called the basis of V * dual to {α s | s ∈ S}. Each configuration f of the lit-only σ-game on Γ is interpreted as the vector if all vertices of Γ are assigned the off state by f, we interpret (2) as the zero vector of V * . Given s ∈ S and f ∈ V * observe that f (α s ) = 1 (resp. 0) if and only if the vertex s is assigned the on (resp. off ) state by f. For each s ∈ S define a linear transformation κ s : V * → V * by Fix a vertex s of Γ. Given any f ∈ V * , if the state of s is on then κ s f is obtained from f by changing the states of all neighbors of s, and κ s f = f otherwise. Therefore we may view κ s as the move of the lit-only σ-game on Γ for which we choose the vertex s and change the states of all neighbors of s if the state of s is on. In particular κ 2 s = 1, the identity map on V * , and so κ s ∈ GL(V * ), the general linear group of V * . The subgroup H = H Γ of GL(V * ) generated by the κ s for all s ∈ S was first mentioned by Wu [18], which is called the flipping group of Γ in [11] and the lit-only group of Γ in [19].
The simply-laced Coxeter group W = W Γ associated with Γ = (S, R) is defined by generators and relations. The generators are the elements of S and the relations are By [11,Theorem 3.2], there exists a unique representation κ = κ Γ : W → GL(V * ) such that κ(s) = κ s for all s ∈ S. Clearly κ(W ) = H. Given any f, g ∈ V * observe that g can be obtained from f by a finite sequence of the moves of the lit-only σ-game on Γ if and only if there exists w ∈ W such that g = κ(w)f. Given an integer k, the underlying graph Γ is k-lit if and only if for each for all s, t ∈ S [15]. The radical of V (relative to B) is the subspace of V consisting of the vectors α that satisfy B(α, β) = 0 for all β ∈ V. The form B is said to be degenerate whenever the radical of V is nonzero and nondegenerate otherwise. The graph Γ is said to be degenerate whenever the form B is degenerate, and nondegenerate otherwise. The form B induces a linear map θ : V → V * given by Since the kernel of θ is the radical of V and the matrix representing B with respect to the basis {α s | s ∈ S} is the adjacency matrix of Γ over F 2 , the following lemma is straightforward.
Lemma 2.1. Let A denote the adjacency matrix of Γ over F 2 . Then the following are equivalent: (i) Γ is a nondegenerate graph.
(ii) θ is an isomorphism of vector spaces.
The purpose of this paper is to investigate the lit-only σ-game on nondegenerate graphs which are not line graphs. It is natural to ask how to determine if a nondegenerate graph is a line graph. Here we give two characterizations of nondegenerate line graphs.

Lemma 2.2.
Assume that Γ is the line graph of a simple connected graph G of order n. Then θ(V ) has dimension n − 1 if n is odd and has dimension n − 2 if n is even.
where u and v are the two endpoints of s in G. Since G is connected, the image of µ is the subspace of U consisting of these vectors each of which equals the sum of an even number of vertices of U. Define a linear map λ : U → V * by for all u ∈ U and for all s ∈ S. There is only one nonzero vector, the sum of all vertices of G, in the kernel of λ. Since θ = λ • µ and by the above comments, the result follows.
A claw is a tree with one internal vertex and three leaves. A simple graph is said to be claw-free if it does not contain a claw as an induced subgraph. A cut-vertex of Γ is a vertex of Γ whose deletion increase the number of components. A block of Γ is a maximal connected subgraph of Γ without cut-vertices. A block graph is a simple connected graph in which every block is a complete graph.  (i) Γ is a nondegenerate line graph.
(ii) Γ is the line graph of an odd-order tree.
(iii) Γ is a claw-free block graph of even order.

Main results
Let GL(V ) denote the general linear group of V. Given a quadratic form Q on V , the orthogonal group with respect to Q is the subgroup of GL(V ) consisting of all σ ∈ GL(V ) such that Q(σα) = Q(α) for all α ∈ V. Given a basis P of V we define Q P to be the quadratic form on V satisfying Q P (α) = 1 for all α ∈ P . For the rest of this paper, the form B is assumed to be nondegenerate. Moreover let Q = Q P where P = {α s | s ∈ S} and let O(V ) denote the orthogonal group with respect to Q. We now state the main results of this paper, which are Theorem 3.1, Theorem 3.2, and Corollary 3.3. Under the assumption that B is nondegenerate, the number |S| = 2m is even and there exists a basis {β 1 , γ 1 , . . . , β m , γ m } of V such that B(β i , β j ) = 0, B(γ i , γ j ) = 0 and which is independent on the choice of the symplectic basis {β 1 , γ 1 , . . . , β m , γ m } of V (for example see [1] or [8,Theorem 13.13]). Any two quadratic forms over for all t ∈ S. The set {α ∨ s | s ∈ S} forms a basis of V and is called the basis of V dual to {α s | s ∈ S} (with respect to B). Theorem 3.2. Assume that Γ = (S, R) is a nondegenerate graph but not a line graph. Then Γ is 2-lit. Moreover the following are equivalent: When the nondegenerate graph Γ is bipartite, Theorem 3.2 can be reduced as follows.
Corollary 3.3. Assume that Γ is a nondegenerate bipartite graph. Then Γ is 2-lit. Moreover the following are equivalent: (i) Γ is 1-lit (ii) Γ contains a vertex with even degree or Γ is a single edge.
As consequences of Corollary 3.3, we obtain two families of 1-lit graphs as follows.
• A tree is nondegenerate if and only if it has a perfect matching. By [10, Lemma 2.4], a tree with a perfect matching satisfies Corollary 3.3(ii) and is therefore 1-lit (cf. [13, Theorem 1.1]).
• For any two positive integers m and n, the m × n grid is nondegenerate if and only if m + 1 and n + 1 are coprime [17]. By Corollary 3.3 any such m × n gird is 1-lit.
The following example shows that Corollary 3.3 is no longer true if the assumption of Γ is the same as that of Theorem 3.2. Consider the graph Γ = (S, R) as below. The graph Γ = (S, R) is nondegenerate and not a block graph. Therefore Γ is not a line graph by Proposition 2.4. The basis {α ∨ 1 , α ∨ 2 , . . . , α ∨ 6 } of V dual to {α 1 , α 2 , . . . , α 6 } can be expressed as follows.
Using (6) and Q(α s ) = 1 for all s ∈ S, we deduce that Q(α ∨ s ) = 0 for all s ∈ S. Therefore Γ is not 1-lit by Theorem 3.2, but the vertices 2, 5 have even degree in Γ.

Proof of Theorem 3.1
For α ∈ V the transvection on V with direction α is a linear transformation τ α : V → V defined by Observe that τ α preserves the form B and that τ α ∈ GL(V ) since τ 2 α = 1. Here 1 denotes the identity map on V.
For a subset P of V define T v(P ) to be the subgroup of GL(V ) generated by τ α for α ∈ P, and define G(P ) to be the simple graph whose vertex set is P and where α, β in P form an edge if and only if B(α, β) = 1. For any two linearly independent sets P and P ′ of V , we say that P ′ is elementary t-equivalent to P whenever there exist α, β ∈ P such that P ′ is obtained from P by changing β to τ α β. The equivalence relation generated by the elementary t-equivalence relation is called the t-equivalence relation [3]. Theorem 3.3]. Let P denote a linearly independent set of V . Assume that G(P ) is a connected graph. Then there exists P ′ in t-equivalence class of P for which G(P ′ ) is a tree. Lemma 3.7]. Let P denote a linearly independent set of V . Assume that G(P ) is the line graph of a tree. Then, for each P ′ in the t-equivalence class of P , the graph G(P ′ ) is the line graph of a tree.
A basis P of V is said to have orthogonal type [4] if P is t-equivalent to some P ′ for which G(P ′ ) is a tree containing the graph as a subgraph. Lemma 4.3. Assume that P is a basis of V for which G(P ) is a tree but not a path. Then P is of orthogonal type.
Proof. Since G(P ) is not a path it contains a vertex α with degree at least three. If any two neighbors of α, say β and γ, are leaves of G(P ), then β + γ lies in the radical of V , which contradicts that B is nondegenerate. Therefore at most one neighbor of α is a leaf in G(P ) and so P is of orthogonal type.
Lemma 4.4. [4, Section 10]. Let P denote a basis of V which is of orthogonal type. Then T v(P ) is the orthogonal group with respect to Q P . Moreover the T v(P )-orbits on V are . Proof of Theorem 3.1. For each s ∈ S let τ s denote the transvection on V with direction α s . By [15,Section 5], there exists a unique representation τ = τ Γ : W → GL(V ) such that τ (s) = τ s for all s ∈ S. For each w ∈ W the transpose of τ (w −1 ) is equal to κ(w). Therefore κ is the dual representation of τ . Since τ preserves the form B we have Let P = {α s | s ∈ S}. Clearly T v(P ) = τ (W ) and G(P ) is (isomorphic to) Γ. By Lemma 4.1 there exists P ′ in t-equivalence class of P for which G(P ′ ) is a tree. Since G(P ) is not a line graph, the tree G(P ′ ) is not a path by  Definition 5.1. We define R ∨ as the set consisting of all two-element subsets {s, t} of S with B(α ∨ s , α ∨ t ) = 1. Define Γ ∨ as the simple graph with vertex set S and edge set R ∨ . We will refer to Γ ∨ as the dual graph of Γ.
Note that the notion of dual graphs defined above is different from the usual ones in graph theory. The following lemma suggests why the graph Γ ∨ is of interest.
Proof. Let s, t ∈ S be given. Using (5) and (7) we have θ(α ∨ s )α t = 1 whenever s = t and otherwise θ(α ∨ s )α t = 0. Comparing this with (1) the result follows. Recall from Section 2 that the symplectic form B is defined on the basis {α s | s ∈ S} of V . If the symplectic form associated with Γ ∨ is defined on the basis {α ∨ s | s ∈ S} of V , then the resulting form is B. Therefore Γ ∨ is a nondegenerate graph. The dual graph of Γ ∨ is Γ since {α s | s ∈ S} is the basis of V dual to {α ∨ s | s ∈ S}.
By duality and Lemma 5.3 the following lemma is straightforward.
Lemma 5.5. Let A and A ∨ denote the adjacency matrices of Γ and Γ ∨ over F 2 , respectively. Then A and A ∨ are inverses of each other.
Proof. We show that A ∨ A is equal to the identity matrix. Let s, t ∈ S be given. By the comment below Lemma 5.2 the (s, t)-entry of A (resp. A ∨ ) is equal to B(α s , α t ) (resp. B(α ∨ s , α ∨ t )). By Definition 5.1 we find that the (s, t)-entry of A ∨ A equals By (11) the vector in the first coordinate of (12) equals α ∨ s . Therefore (12) equals 1 if and only if s = t by (7). The result follows.
We are now ready to prove Theorem 3.2.
Proof of Theorem 3.2. In Lemma 5.2 we saw that θ(α ∨ s ) = f s for all s ∈ S. Therefore (i), (ii) are equivalent by Theorem 3.1. To show that Γ is 2-lit, it is now enough to consider the two cases: (a) Q(α ∨ s ) = 0 for all s ∈ S; (b) Q(α ∨ s ) = 1 for all s ∈ S. (a) It suffices to show that there exist s, t ∈ S such that Q(α ∨ s + α ∨ t ) = 1. Since the form B is nontrivial there exist s, t ∈ S such that B(α ∨ s , α ∨ t ) = 1. Then the s and t are the desired elements in S.
(b) It suffices to show that there exist two distinct s, t ∈ S such that Q(α ∨ s + α ∨ t ) = 0. By our assumption, the graph Γ is not a complete graph. Using Lemma 5.5, we deduce that Γ ∨ is not a complete graph. Therefore there exist two distinct s, t ∈ S such that B(α ∨ s , α ∨ t ) = 0. Such s and t are desired elements in S.
To prove Corollary 3.3, we give a sufficient condition for Theorem 3.2(ii).
Lemma 5.6. Let Γ = (S, R) denote a nondegenerate graph. Assume that there exists s ∈ S with even degree in Γ such that where the sum is over all two-element subsets {u, v} of S with su, sv ∈ R. Then the restriction of Q to {α ∨ t | st ∈ R} is surjective. Proof. Apply Q to either side of (9). Using (6), (13) and Q(α s ) = 1 to evaluate the resulting equation, we obtain that st∈R Q(α ∨ t ) = 1.
By (14) there exists a neighbor u of s for which Q(α ∨ u ) = 1. Since s has even degree in Γ there exists a neighbor v of s for which Q(α ∨ v ) = 0. The result follows.
Proof of Corollary 3.3. By Proposition 2.4 a nondegenerate bipartite graph Γ is a line graph if and only if Γ is a path of even order. Since every path is 1-lit, this corollary holds for Γ as a line graph. We thus assume that Γ is not a line graph. By Theorem 3.2 the graph Γ is 2-lit. By Lemma 5.5 we deduce that the graph Γ ∨ is bipartite with bipartition as same as that of Γ. We use this to show that (i), (ii) are equivalent.
(ii) ⇒ (i): Let s denote a vertex of Γ with even degree. Since Γ and Γ ∨ are bipartite graphs with same bipartition, we deduce that B(α ∨ u , α ∨ v ) = 0 for any neighbors u, v of s in Γ. Therefore (13) holds. By Lemma 5.6 the restriction of Q on {α ∨ t | st ∈ R} is onto. Therefore Γ is 1-lit by Theorem 3.2.
(i) ⇒ (ii): Suppose on the contrary that each vertex of Γ has odd degree. Using Lemma 5.5 we deduce that each vertex of Γ ∨ has odd degree. Let s denote any element of S. By (11) the value Q(α ∨ s ) equals Since the bipartite graphs Γ and Γ ∨ have the same bipartition, we deduce that B(α u , α v ) = 0 for any neighbors u, v of s in Γ ∨ . By (6) the summation in (15) can be moved out front.