Minimum Light Numbers in the σ-Game and Lit-Only σ-Game on Unicyclic and Grid Graphs

Consider a graph each of whose vertices is either in the ON state or in the OFF state and call the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph. A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a regular move at an ON vertex. For any graph G, let D(G) be the minimum integer such that given any starting configuration x of G there must exist a sequence of valid moves which takes x to a configuration with at most l + D(G) ON vertices provided there is a sequence of regular moves which brings x to a configuration in which there are l ON vertices. The shadow graph S(G) of a graph G is obtained from G by deleting all loops. We prove that D(G) ≤ 3 if S(G) is unicyclic and give an example to show that the bound 3 is tight. We also prove that D(G) ≤ 2 if G is a two-dimensional grid graph and D(G) = 0 if S(G) is a two-dimensional grid graph but not a path and G 6= S(G). 1 Definitions and background A graph G is a pair of sets consisting of its vertex set V (G) and its edge set E(G) such that E(G) ⊆ ( V (G) 2 ) ∪ V (G) and we say that there is a loop at a vertex v of G provided {v} ∈ E(G). When u 6= v, we often denote an edge {u, v} by uv; a singleton set {u} is mostly just written as u. Two different vertices u and v of G are adjacent provided uv ∈ E(G) and v is adjacent to itself if v ∈ V (G) ∩ E(G) (so there is a loop at v). The set of vertices adjacent to a given vertex v in G is designated by NG(v) and called the Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA. Email: [email protected]. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. Email: [email protected]. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China. Email: [email protected]. Fax: 86-21-54743152. Corresponding author. the electronic journal of combinatorics 18 (2011), #P214 1 set of neighbors of v in G. For S ⊆ V (G), we put G[S] to be the graph with vertex set S and edge set E(G) ∩ ( ( S 2 ) ∪ S). For any v ∈ V (G), we use the abbreviation G− v for G[V (G) \ {v}]. The degree of a vertex v in a graph G is defined to be the number of edges in E(G) \ V (G) that contain v and we will use the notation degG(v) for it. We say that v is a branch vertex of G if degG(v) ≥ 3. For any k positive integers m1, . . . , mk, the k-dimensional grid graph Gm1,...,mk has vertex set {vi1,...,ik : 1 ≤ i1 ≤ m1, . . . , 1 ≤ ik ≤ mk} and {vi1,...,ik , vj1,...,jk} ∈ E(Gm1,...,mk) if and only if ∑k t=1(it − jt) 2 = 1. The graph Gn is often called an n-path and denoted [v1, . . . , vn]; see Fig. 1. For any n ≥ 3, the graph obtained from the path [v1, . . . , vn] by adding an edge v1vn is referred to as an n-cycle and is denoted 〈v1, . . . , vn〉. Note that a 4-cycle is nothing but G2,2. A unicyclic graph is a loopless connected graph containing exactly one cycle. s s s s s v1 v2 . . . vn−1 vn Figure 1: An n-path [v1, . . . , vn]. The shadow graph of a graph G, which we denote by S(G), is the (loopless) graph with vertex set V (G) and edge set E(G) \ V (G). If S(G) is a tree, we call G a pseudo-tree. Similarly, we can talk about a pseudo-cycle and a pseudo-unicyclic graph, etc.. Let F2 be the binary field and we refer to any element x of F V (G) 2 as a configuration of G. We say that v ∈ V (G) is ON in x if x(v) = 1 and is OFF in x if x(v) = 0 and hence we can also regard a configuration as an assignment of states ON (1) or OFF (0) to vertices of G, or simply an ordered bipartition of V (G) into ON vertex set and OFF vertex set. The light number L(x) of a configuration x is the number of ON vertices in x, namely L(x) = |supp(x)|, where supp(x) means the support of the function x. For any x ∈ F V (G) 2 and any U ⊆ V (G), xU is the restriction of x on U, namely the image of x under the natural projection from F V (G) 2 to F U 2 . For any S ⊆ V (G), χS ∈ F V (G) 2 is the characteristic vector of S. Note that χS + χQ = χS△Q where △ stands for taking the symmetric difference of two sets. We will write χ{v} simply as χv. It is clear that each configuration x is just the characteristic vector of supp(x). In a graphical representation of a configuration, we often use a circle for a vertex in the OFF state and a bullet for a vertex in the ON state; see Fig. 2 for an example.