Abstract
Graph Theory Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.
Highlights
Bonomo, Figueiredo, Duran, Grippo, Safe, and SzwarcfiterLet G be a class of graphs
We investigate probe 2-clique graphs and probe diamond-free graphs
A graph G = (V, E) is probe G if its vertices can be partitioned into probe vertices (P ) and a stable set of nonprobe vertices (N ) in such a way that there exists a set of non-edges F of G, whose endpoints belong to N, such that G∗ = (V, E ∪ F ) is in G
Summary
Given a graph a G with a prescribed partition (P, N ) of its vertex set into probe (P) and nonprobe (N) vertices, the problem of decided whether there exists a completion of G = (P ∪ N, E) into an interval graph G∗ = (P ∪ N, E ∪ F ) can be solved in O(n2)-time [13], this algorithm involves modified PQtrees. Another algorithm, which deals with this problem, is the one presented in [16] that uses modular decomposition and whose complexity is O(n + m log(n)). We conclude the article with a section of further remarks
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