Recognizing simple-triangle graphs by restricted 2-chain subgraph cover

A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was a longstanding open problem, and recently a polynomial-time algorithm has been given (Mertzios, 2015). Along with the approach of this paper, we show a simpler recognition algorithm for simple-triangle graphs. To do this, we introduce the following problem: given a bipartite graph G and a set F of edges of G, find a 2-chain subgraph cover of G such that one of two chain subgraphs has no edges in F. We show an O(n(2) . min{m, (m) over cap})-time algorithm for this problem, where n and m are the number of vertices and edges of G, respectively, and (m) over cap is the number of non-edges between two independent sets of G. Using this result, we show that the recognition problem for simple-triangle graphs can be solved in O(n(2) . min{m, (m) over bar}) time, where (m) over bar is the number of non-edges of the given graph. (C) 2019 Elsevier B.V. All rights reserved.