Quasiperfect domination in triangular lattices

A vertex subset $S$ of a graph $G$ is a perfect (resp. quasiperfect) dominating set in $G$ if each vertex $v$ of $G\setminus S$ is adjacent to only one vertex ($d_v\in\{1,2\}$ vertices) of $S$. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schl\"afli symbol $\{3,6\}$ and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets $S$ with induced components of the form $K_{\nu}$, where $\nu\in\{1,2,3\}$ depends only on $S$.Comment: 20 pages, 9 figures, 5 array

On perfect and quasiperfect dominations in graphs

A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Postprint (published version

On perfect and quasiperfect domination in graphs

Preprin

EFFICIENT AND PERFECT DOMINATION ON ARCHIMEDEAN LATTICES

An Archimedean lattice is an infinite graph constructed from a vertex-transitive tiling of the plane by regular polygons. A set of vertices S is said to dominate a graph G=(V,E) if every vertex in V is either in the set S or is adjacent to a vertex in set S. A dominating set is a perfect dominating set if every vertex not in the dominating set is dominated exactly once. The domination ratio is the minimum proportion of vertices in a dominating set. The perfect domination ratio is the minimum proportion of vertices in a perfect dominating set. Dominating sets are provided to establish upper bounds for the domination ratios of all the Archimedean lattices. A dominating set is an efficient dominating set if every vertex is dominated exactly once. We show that seven of the eleven Archimedean lattices are efficiently dominated, which easily determine their domination ratios and perfect domination ratios. We prove that the other four Archimedean lattices cannot be efficiently dominated. For the four Archimedean lattices that cannot be efficiently dominated, we have determined their exact perfect domination ratios. Integer programming bounds for domination ratios are provided. A perfect domination proportion is the proportion of vertices in a perfect dominating set that is not necessarily minimal. We study nonisomorphic perfect dominating sets and possible perfect domination proportions of Archimedean lattices