Planar Median Graphs and CubesquareGraphs
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 4cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquaregraphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4cycle boundaries (relative to some embedding) by cubes or squaregraphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and squaregraphs by a sequence of ``squareboundary'' 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 squaregraphs.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09346
 Bibcode:
 2021arXiv211009346S
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics