A co-bipartite chain graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cut problem (MaxCut) is NP-hard in co-bipartite graphs [3]. We consider MaxCut in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that MaxCut is polynomial time solvable in this graph class.
Abstract. The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and NonSplitting Paths in a Tree (ENPT) [2]. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph class by allowing the host graph to be an arbitrary graph. These are the graphs of EdgeIntersecting and Non-Splitting Paths ENP. Although the Edge Intersection Graphs of Paths in an arbitrary graph includes all graphs, we show that this is not the case for ENP. We also show that the class ENP coincides with the family of graphs of Edge-Intersecting and Non-Splitting Paths in a Grid (ENPG). Following similar studies for EPG graph class, we study the implications of restricting the number of bends in the grid, of the individual paths. We show that restricting the bend number also restricts the graph class. Specifically, by restricting the number of bends one gets an infinite sequence of classes such that every class is properly included in the next one.
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