Linear-time algorithms for three domination-based separation problems in block graphs

Abstract

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view, even for several graph classes where other in general hard problems are easy to solve, like bipartite graphs or chordal graphs. Hence, a typical line of attack for these problems is to determine minimum codes of special graphs. In this work we study the problem of determining the cardinality of minimum such codes in block graphs (that are diamond-free chordal graphs). We present linear-time algorithms for these problems, as a generalization of a linear-time algorithm proposed by Auger in 2010 for identifying codes in trees. Thereby, we provide a subclass of chordal graphs for which all three problems can be solved in linear time.

On three domination numbers in block graphs

“…The three here studied domination problems are challenging both from a theoretical and a computational point of view and even remain hard for several graph classes where other in general hard problems are easy to solve, including bipartite graphs and chordal graphs. Block graphs form a subclass of chordal graphs for which all three domination problems can be solved in linear time [1]. In this paper, we complement this result by presenting for all three codes lower and upper bounds.…”

“…In particular, any tree is a block graph. Linear-time algorithms to compute all three domination numbers in block graphs are presented in [1], thus block graphs are chordal graphs for which the three problems can be solved in linear time.…”