Hardness and efficiency on [formula omitted]-admissibility for graph operations

Abstract

The t-admissibility problem aims to decide whether a graph G has a spanning tree T in which the distance between any two adjacent vertices of G is at most t. In this case, G is called t-admissible, and the smallest t for which G is t-admissible is the stretch index of G. A complementary prism of G, denoted by GG¯, is obtained by the union of G with its complement G¯ and the addition of a perfect matching between corresponding vertices of G and G¯. One of the challenges of the t-admissibility problem is to determine 3-admissible graph classes, since the computational complexity of such a problem remains open for more than 25 years. Moreover, it is known that recognizing 4-admissible graphs is, in general, an NP-complete problem (Cai and Corneil, 1995), as well as recognizing t-admissible graphs for graphs with diameter at most t+1, for t≥4 (Papoutsakis, 2013). We prove that any graph G, non-complete graph, can be transformed into a 4-admissible one, by obtaining GG¯. Furthermore, we prove that the stretch indexes of GG¯ graphs are equal to 4, and since they have diameter at most t+1, we present a class for which t-admissibility is solved in polynomial time. GG graphs are the Cartesian product G×K2, defined by the union of two copies of G and the addition of a perfect matching between corresponding vertices of the two graphs G. Interestingly, we prove that for GG graphs, whose definition is very similar to GG¯’s, t-admissibility is NP-complete. Generalizing these constructions, we prove that determining t-admissibility is NP-complete for graphs that have perfect matching.