The matching book embedding of the $F$-sum of two graphs

TL;DR

This paper introduces the concept of matching book embedding and matching book thickness, analyzing dispersability of outerplanar graphs and bounds for the $F$-sum of graphs.

Contribution

It defines matching book embedding, characterizes dispersable graphs, and establishes bounds on matching book thickness for the $F$-sum of graphs.

Findings

Dispersability of outerplanar graphs is characterized.

An upper bound on matching book thickness for the $F$-sum is established.

Matching book thickness relates to maximum degree and graph operations.

Abstract

The $F$-sum is a new graph operation defined by combining four graph transformation operations with the Cartesian product operation. A matching book embedding of a graph $G$ is a book embedding in which the vertices of $G$ are placed on a fixed linear order along the spine, and the edges are assigned to pages such that (i) no two edges on the same page cross, and (ii) each vertex has degree at most one on every page. The minimum number of pages required for such a matching book embedding is called the \emph{matching book thickness} of $G$, denoted by $mbt(G)$. A graph $G $ is dispersable if and only if $ mbt(G) = \Delta(G) $, and nearly dispersable if and only if $mbt(G) = \Delta(G) + 1 $. In this paper, we determine the dispersability of outerplanar graphs and establish an upper bound on the matching book thickness of the $F$-sum of any simple graph with any dispersable bipartite graph.