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The localization game is a variant of the game of Cops and Robber in which the robber is invisible and moves between adjacent vertices, but the cops can probe any ▫$k$▫ vertices of the graph to obtain the distance between probed vertices and the robber. The localization number of a graph is the mini ...

Množica vozlišč ▫$X\subseteq V(G)$▫ je ▫$d$▫-razdaljna dominantna množica, če za vsak ▫$u\in V(G)\setminus X$▫ obstaja ▫$x\in X$▫, tako da je ▫$d(u,x) \le d$▫, in ▫$X$▫ je ▫$p$▫-pakiranje, če je ▫$d(u,v) \ge p+1$▫ za vsaka različna ▫$u,v\in X$▫. Za graf ▫$G$▫ je ▫$d$▫-razdaljno ▫$p$▫-pakirno število ...

Motivated by the burning and cooling processes, the burning game is introduced. Two players (Burner and Staller) play the game on a graph ▫$G$▫ by alternately selecting vertices of ▫$G$▫ to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all v ...

Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order n is at most ⌈1/3n⌉, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture is false in general, but Henning and Dorbec showed that it holds for graphs with ...