Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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.  prove that D(G) ≤ 3 if S(G) is unicyclic and give an example to show that the bound 3 is tight.  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).
['John L. Goldwasser', 'Xinmao Wang', 'Yaokun Wu']
I am writing a research paper on a idea Minimum light numbers in the σ-game and lit-only σ-game on unicyclic and grid graphs
[{"from": "human", "value": "I am writing a research paper on a idea Minimum light numbers in the \u03c3-game and lit-only \u03c3-game on unicyclic and grid graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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.  prove that D(G) \u2264 3 if S(G) is unicyclic and give an example to show that the bound 3 is tight.  also prove that D(G) \u2264 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)."}]