Graphs of Edge-Intersecting and Non-Splitting One Bend Paths in a Grid

TL;DR

This paper studies one bend edge-intersecting path graphs in grids, characterizes their properties, and provides recognition algorithms, revealing their complexity and relationships with other graph classes.

Contribution

It characterizes one bend ENPG graphs, compares them with two bend graphs, and offers recognition algorithms and complexity results.

Findings

One bend ENPG graphs are properly included in two bend ENPG graphs.

Trees and cycles are one bend ENPG graphs.

Recognition of one bend ENPG split graphs is NP-complete, while for co-bipartite graphs it is linear time.

Abstract

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by this result, in this work we focus on one bend {ENPG} graphs. We show that one bend ENPG graphs are properly included in two bend ENPG graphs. We also show that trees and cycles are one bend ENPG graphs, and characterize the split graphs and co-bipartite graphs that are one bend ENPG. We prove that the recognition problem of one bend ENPG split graphs is NP-complete even in a very restricted subfamily of split graphs. Last we provide a linear time recognition algorithm for one bend ENPG co-bipartite graphs.