Planar Median Graphs and Cubesquare-Graphs

TL;DR

This paper characterizes planar median graphs through forbidden subgraphs, cycle structures, and a new construction method using cubes and square-graphs, leading to an efficient recognition algorithm.

Contribution

It introduces novel characterizations of planar median graphs and a new construction framework using cubes and square-graphs, along with an efficient recognition algorithm.

Findings

Characterization of planar median graphs via forbidden subgraphs and cycle structures.

Introduction of cubesquare-graphs as a construction method for planar median graphs.

Development of an $ O(n \log n)$-time recognition algorithm.

Abstract

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.