Edge covering pseudo-outerplanar graphs with forests

TL;DR

This paper studies pseudo-outerplanar graphs, proving they can be decomposed into various combinations of forests, matchings, and outerplanar graphs, extending known results for outerplanar and $K_{2,3}$-minor-free graphs.

Contribution

It establishes new edge decomposition results for pseudo-outerplanar graphs, broadening the scope of previous work on related graph classes.

Findings

Decomposition into a linear forest and an outerplanar graph

Decomposition into a star forest and an outerplanar graph

Decomposition into two forests and a matching

Abstract

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or two forests and a matching, or $\max\{\Delta(G),4\}$ matchings, or $\max\{\lceil\Delta(G)/2\rceil,3\}$ linear forests. These results generalize some ones on outerplanar graphs and $K_{2,3}$-minor-free graphs, since the class of pseudo-outerplanar graphs is a larger class than the one of $K_{2,3}$-minor-free graphs.