Lit-only sigma-game on nondegenerate graphs

A configuration of the lit-only σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-game on a graph Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} is an assignment of one of two states, on or off, to each vertex of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}. Given a configuration, a move of the lit-only σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-game on Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} allows the player to choose an on vertex s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s$$\end{document} of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} and change the states of all neighbors of s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s$$\end{document}. Given an integer k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}, the underlying graph Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} is said to be k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-lit if for any configuration, the number of on vertices can be reduced to at most k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} by a finite sequence of moves. We give a description of the orbits of the lit-only σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-game on nondegenerate graphs Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} which are not line graphs. We show that these graphs Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} are 2-lit and provide a linear algebraic criterion for Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} to be 1-lit.


Introduction
The notion of the σ -game on finite graphs was first introduced by Sutner [17,18] around 1989. A configuration of the σ -game on is an assignment of one of two states, on or off, to each vertex of . Given a configuration, a move consists of choosing a vertex of , followed by changing the states of all of its neighbors. If only on vertices can be chosen 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 is to minimize the number of on vertices of , or to reach an assigned configuration by a finite sequence of moves.
H. Huang (B) Mathematics Division, National Center for Theoretical Sciences, National Tsing-Hua University, Hsinchu 30013, Taiwan, ROC e-mail: hauwenh@math.cts.nthu.edu.tw Given an integer k, the underlying graph 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 moves. More precisely, we are interested in the orbits of the lit-only σ -game on and the smallest integer k, the minimum light number of [19], for which 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 [19], where the name "lit-only σ -game" was coined.
The lit-only σ -game on a simple graph is simply the natural action of a certain subgroup H of the general linear group over F 2 [19]. Under the assumption that is the line graph of a simple graph G, Wu [21] described the orbits of the lit-only σ -game on and gave a characterization for the minimum light number of . Moreover, if G is a tree of order n ≥ 3, Wu showed that H is isomorphic to the symmetric group on n letters. Weng and the author [13] determined the structure of H without any assumption on G. The lit-only σ -game on a simple graph can also be considered as a representation κ of the simply-laced Coxeter group W over F 2 [12]. The dual representation of κ preserves a certain symplectic form B . The two representations are equivalent whenever the form B is nondegenerate. From this viewpoint it is natural to partition simple connected graphs into two classes according as B is degenerate or nondegenerate.
In this paper, we treat nondegenerate graphs which are not line graphs. We show that H is isomorphic to an orthogonal group, followed by a description of the orbits of lit-only σ -game on (Theorem 3.1). Moreover, we show that these graphs are 2-lit and provide a linear algebraic criterion for to be 1-lit (Theorem 3.2). Combining Theorem 3.1, Theorem 3.2, and those in [13,21], 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 = (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 an F 2 -vector space that has a basis {α s | s ∈ S} in one-to-one correspondence with S.
If all vertices of 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 . 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 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. For any vector space U , let GL(U ) denote the general linear group of U . Then κ s ∈ GL(V * ) for all s ∈ S. The subgroup H = H of GL(V * ) generated by the κ s for all s ∈ S was first mentioned by Wu [19], which is called the flipping group of in [12] and the lit-only group of in [21]. The lit-only groups are closely related to the simply-laced Coxeter groups in the following way. Recall that the simply-laced Coxeter group W = W associated with = (S, R) is the group generated by all elements s ∈ S subject to the relations for all s, t ∈ S. By [12, Theorem 3.2], there exists a unique representation κ = κ : 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 moves of the lit-only σ -game on if and only if there exists w ∈ W such that g = κ(w) f . Given an integer k, the underlying graph is k-lit if and only if for each κ(W )-orbit O on V * , there exists a subset K of S with size at most k such that s∈K f s ∈ O.
We now give the definitions of degenerate and nondegenerate graphs. Let B = B denote the symplectic form on V defined by for all s, t ∈ S [16]. 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 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 over F 2 , the following lemma is straightforward.

Lemma 2.1
Let A denote the adjacency matrix of over F 2 . Then the following are equivalent: Recall that given a simple graph G, the line graph of G is a simple graph that has a vertex for each edge of G, and two of these vertices are adjacent whenever the corresponding edges in G have a common vertex. The purpose of this paper is to investigate the lit-only σ -game on nondegenerate graphs which are not line graphs. Thus, it is natural to ask how to determine if a nondegenerate graph is a line graph. We will give two characterizations of nondegenerate line graphs as Proposition 2.4 below.

Lemma 2.2 Let G denote a finite simple connected graph of order n. Assume that is the line graph of G. Then θ(V ) has dimension n − 1 if n is odd and has
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 : 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 is a vertex of whose deletion increases the number of components. A block of is a maximal connected subgraph of without cut-vertices. A block graph is a simple connected graph in which every block is a complete graph.

Lemma 2.3 [10, Theorem 8.5]. Let denote a simple connected graph. Then is the line graph of a tree if and only if is a claw-free block graph.
The following proposition follows by combining Lemmas 2.1-2.3.

Proposition 2.4 Let denote a simple connected graph. Then the following are equivalent: (i) is a nondegenerate line graph. (ii) is the line graph of an odd-order tree.
(iii) is a claw-free block graph of even order.

Main results
A quadratic form Q on V is a function Q : V → F 2 satisfying 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(σ α) = Q(α) for all α ∈ V . Given a basis P of V we define Q P to be the unique quadratic form on V with 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. By (6), for any T ⊆ S a combinatorial interpretation of Q s∈T α s is the parity of the number of vertices and edges on the subgraph of induced by T .
We now can 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 which is independent of the choice of the symplectic basis {β 1 , γ 1 , . . . , β m , γ m } of V (for example see [1] 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 = (S, R) is a nondegenerate graph, but not a line graph.
Then is 2-lit. Moreover, the following are equivalent: When the nondegenerate graph is bipartite, Theorem 3.2 can be improved as follows.

Corollary 3.3 Assume that is a nondegenerate bipartite graph. Then is 2-lit. Moreover, the following are equivalent: (i) is 1-lit (ii) contains a vertex with even degree or is a single edge.
As consequences of Corollary 3.3, we obtain two families of 1-lit graphs as follows. if m + 1 and n + 1 are coprime [18]. By Corollary 3.3 any such m × n grid is 1-lit. This result partially improves [8,Theorem 26].
The following example shows that Corollary 3.3 is no longer true if the assumption of is the same as that of Theorem 3.2. Consider the graph = (S, R) as below. The graph = (S, R) is nondegenerate and not a block graph. Therefore 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.
A direct computation shows that Q(α ∨ s ) = 0 for all s ∈ S. Therefore is not 1-lit by Theorem 3.2, but the vertices 2,5 have even degree in .

Proof of Theorem 3.1
To prove Theorem 3.1, we consider a family of linear transformations on V defined as follows. 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. 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].

Lemma 4.1 [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 4.2 [15,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
Proof of Theorem 3.1. For each s ∈ S, let τ s denote the transvection on V with direction α s . By [16,Section 5], there exists a unique representation τ = τ : 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) . By Applying (8) and since θ is an isomorphism by Lemma 2.1, the result follows.

Proof of Theorem 3.2 and Corollary 3.3
Recall the basis {α ∨ s | s ∈ S} of V from (7). To prove Theorem 3.2 and Corollary 3.3, we introduce a simple graph which includes the information of the values B(α ∨ s , α ∨ t ) for all s, t ∈ S as follows.
Define R ∨ to be the set consisting of all two-element subsets {s, t} of S with B(α ∨ s , α ∨ t ) = 1. Define ∨ to be the simple graph with vertex set S and edge set R ∨ . We will refer to ∨ as the dual graph of . 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 ∨ is of interest.

Lemma 5.2 For each s ∈ S we have
Proof Fix s ∈ S. By (1), (4), and (5) By Lemma 5.3 the vector in the first coordinate of (9) is equal to α ∨ s . Therefore (9) is equal to 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 5.1 we saw that θ(α ∨ s ) = f s for all s ∈ S. Therefore (i) and (ii) are equivalent by Theorem 3.1. To show that 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 is not a complete graph. Using Lemma 5.4, we deduce that ∨ 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 the desired elements in S.
To prove Corollary 3.3, we give a sufficient condition for Theorem 3.2(ii).
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 the equation in Lemma 5.2. Using (6), (10) and Q(α s ) = 1 to evaluate the resulting equation, we obtain that st∈R Q(α ∨ t ) = 1.
By (11) there exists a neighbor u of s for which Q(α ∨ u ) = 1. Since s has even degree in 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 is a line graph if and only if is a path of even order. Since every path is 1-lit, this corollary holds for as a line graph. We thus assume that is not a line graph. By Theorem 3.2 the graph is 2-lit. By Lemma 5.4 we deduce that the graph ∨ is bipartite with the same bipartition that of . We use this to show that (i) and (ii) are equivalent.
(ii) ⇒ (i): Let s denote a vertex of with even degree. Since and ∨ are bipartite graphs with same bipartition, we deduce that B(α ∨ u , α ∨ v ) = 0 for any neighbors u, v of s in . Therefore (10) holds. By Lemma 5.5 the restriction of Q on {α ∨ t | st ∈ R} is onto. Therefore is 1-lit by Theorem 3.2. (i) ⇒ (ii): Suppose on the contrary that each vertex of has odd degree. Using Lemma 5.4, we deduce that each vertex of ∨ has odd degree. Let s denote any element of S. By Lemma 5.3, Q(α ∨ s ) is equal to Since the bipartite graphs and ∨ have the same bipartition, we deduce that B(α u , α v ) = 0 for any neighbors u, v of s in ∨ . By (6), the summation in (12) can be moved out front. Since Q(α s ) = 1 for all s ∈ S, it follows that (12) is equal to 1, contradicting Theorem 3.2(ii).