Abstract
Graph Theory A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.
Highlights
Bonomo, Duran, Grippo, and SafeA graph G is an interval graph if it is the intersection graph of a family of open intervals in the real line; i.e., there is a one-to-one correspondence between the vertex set of G and a collection S of intervals in the real line such that two vertices of G are adjacent if their corresponding intervals intersect
An open question from a combinatorial viewpoint is that of characterizing the classes of probe interval and probe unit interval graphs by forbidden induced subgraphs
They present a conjecture describing the complete family of forbidden induced subgraphs for the class of bipartite probe interval graphs
Summary
A graph G is an interval graph if it is the intersection graph of a family of open intervals in the real line; i.e., there is a one-to-one correspondence between the vertex set of G and a collection S of intervals in the real line such that two vertices of G are adjacent if their corresponding intervals intersect. An open question from a combinatorial viewpoint is that of characterizing the classes of probe interval and probe unit interval graphs by forbidden induced subgraphs Partial results in this direction were obtained in the following articles. The main results of this paper are characterizations of probe interval graph and probe unit interval graphs by a set of minimal forbidden induced subgraphs within two superclasses of cographs: treecographs and P4-tidy graphs. Both superclasses of cographs are defined recursively from the disjoint union and the join operation.
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