Complexity results on open-independent, open-locating-dominating sets in complementary prism graphs

Abstract

For a finite, simple, undirected graph G = ( V ( G ) , E ( G ) ) , an open-dominating set S ⊆ V ( G ) is such that every vertex in G has at least one neighbor in S . An open-independent, open-locating-dominating set S ⊆ V ( G ) ( O L D o i n d -set for short) is such that no two vertices in G have the same set of neighbors in S and each vertex in S is open-dominated exactly once by S . The problem of deciding whether or not G has such set is known to be NP -complete. The complementary prism of G is the graph G G ¯ , formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between the corresponding vertices of G and G ¯ . Various properties, upper bounds and logarithmic lower bounds on the sizes of minimal O L D o i n d -sets in complementary prisms are presented. We show that deciding, for a given graph G , whether or not G G ¯ has an O L D o i n d -set is an NP -complete problem. On the other hand, if the girth of G is at least four, we show that the O L D o i n d -set of G G ¯ , if it exists, can be found in polynomial time.