A characterization of always solvable trees in Lights Out game using the activation numbers of vertices

Abstract

Lights out is a game that can be played on any simple graph G. A configuration assigns one of the two states on or off to each vertex. For a given configuration, the aim of the game is to turn all vertices off by applying a push pattern on vertices, where each push switches the state of the vertex and its neighbors. If every configuration of vertices is solvable, then we say that the graph is always solvable. We introduce a concept which we call the activation numbers of vertices and we prove several characterization results of graphs by using this concept. We show that for every always solvable graph there exists a chain of always solvable subgraphs where each subgraph differs from the preceding one by a vertex. We also characterize always solvable trees by showing that all always solvable trees can be constructed from always solvable subtrees by some special types of connections. We call the dimension of the space of null-patterns, which leave configurations unchanged, the nullity of the graph G. We show that the nullity of a tree can be characterized by the cardinality of its minimal partition into always solvable subtrees.

Effects of edge addition or removal on the nullity of a graph in the Lights Out game

“…Remark 2.4. Definitions of half activated, always activated and never activated vertices coincide with the definitions in [5]. Indeed, by ( O 3) we see that, for a vertex u, {u} is not solvable if and only if ℓ(u) = x {u} • ℓ = 1 for some null pattern ℓ.…”

“…How the nullity of a graph change when a vertex is removed or what the nullity of the emergent graph becomes when two graphs are joined together by a single or multiple edges was investigated by several authors [3], [10], [8], [4], [5]. However, as far as we know, until now there has been no study which investigates how nullity changes when an edge is added to or removed from a graph.…”

“…These results enable us to obtain important corollaries such as a characterization of always solvable graphs as we stated in Theorem 3.9. Until now, characterizations of always solvable trees were only known [1], [3], [5]. Therefore, Theorem 3.9 gains importance by being the first example of a characterization of always solvable graphs.…”

Parity of the cardinality of an odd dominating set

“…We form the join graph H := G 1 uwG 2 by connecting the vertices u of G 1 and w of G 2 by an edge. In [ [10], Theorem 3.3], we stated what the nullity of H and the activation numbers of u and w with respect to H become when we join two graphs with specific activation numbers of u and w with respect to G 1 and G 2 , respectively. Moreover, we explained the structure of the component sets of any odd dominating set of H in graphs G 1 and G 2 .…”

“…It turns out that a vertex u is in all, exactly half, or none of the odd dominating sets (see [[3], Lemma 1] and [ [10], Lemma 2.4]. We say u is always, half, or never activated if the first, the second, or the third case holds, respectively.…”