Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

TL;DR

This paper studies monotonic edge intersection graph representations of outerplanar graphs on a grid, establishing an upper bound on their bend number and characterizing graphs with minimal bends, along with efficient construction algorithms.

Contribution

It proves that all outerplanar graphs have a monotonic bend number at most 2 and characterizes those with bend numbers 0, 1, and 2 using forbidden subgraphs.

Findings

Monotonic bend number of outerplanar graphs is at most 2.

Characterization of graphs with bend number 0, 1, and 2.

Polynomial time algorithms for constructing minimal bend representations.

Abstract

In a representation of a graph $G$ as an edge intersection graph of paths on a grid (EPG) every vertex of $G$ is represented by a path on a grid and two paths share a grid edge iff the corresponding vertices are adjacent. In a monotonic EPG representation every path on the grid is ascending in both rows and columns. In a (monotonic) $B_k$-EPG representation every path on the grid has at most $k$ bends. The (monotonic) bend number $b(G)$ ($b^m(G)$) of a graph $G$ is the smallest natural number $k$ for which there exists a (monotonic) $B_k$-EPG representation of $G$. In this paper we deal with the monotonic bend number of outerplanar graphs and show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$ in terms of forbidden induced subgraphs. As a byproduct we obtain low-degree polynomial time algorithms to construct (monotonic) EPG representations with the smallest possible number of bends for maximal outerplanar graphs and cacti.