Hardness and efficiency on t-admissibility for graph operations

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 G (G) over bar, is obtained by the union of G with its complement (G) over bar and the addition of a perfect matching between corresponding vertices of G and (G) over bar. 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 G (G) over bar. Furthermore, we prove that the stretch indexes of G (G) over bar 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 x K-2, 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 G (G) over bar 's, t-admissibility is NP-complete. Generalizing these constructions, we prove that determining t-admissibility is NP-complete for graphs that have perfect matching. (C) 2021 Elsevier B.V. All rights reserved.