Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points V from the circle; and make a graph with vertex set V in which two vertices are adjacent if they both belong to one of the intervals. Such graphs are “circular interval graphs”, and they form an important subclass of the class of all cla...

The edge clique graph of a graph H is the one having the edge set of H as vertex set, two vertices being adjacent if and only if the corresponding edges belong to a common complete subgraph of H. We characterize the graph classes {edge clique graphs} ∩ {interval graphs} as well as {edge clique graphs} ∩ {probe interval graphs}, which leads to polynomial time recognition algorithms for them. Thi...

A graph G is a star-k-PCG if there exists non-negative edge weighted star tree S and k mutually exclusive intervals $$I_1, I_2, \ldots , I_k$$ of reals such that each vertex corresponds to leaf an between two vertices in the distance their corresponding leaves lies $$I_1\cup I_2\cup \cup . These graphs are related different well-studied classes as PCGs multithreshold graphs. It well known for a...

A set of vertices W a graph G is total k-dominating when every vertex has at least k neighbors in W. In recent article, Chiarelli et al. (2019) prove that can be computed O(n3k) time proper interval with n and m edges. this note we reduce the complexity to O(m) for k=2.