Three new decompositions of graphs based on a vertex-removing synchronised graph product

TL;DR

This paper introduces three new graph decomposition theorems based on a vertex-removing synchronised product, extending existing methods to facilitate prime-graph decomposition with applications in synchronising real-time processes.

Contribution

The paper presents three novel graph-decomposition theorems utilizing a modified product, expanding the toolkit for graph analysis and prime-graph decomposition.

Findings

Decomposition based on semicomplete bipartite subgraphs

Introduction of matrix graph for decomposition

Combined decomposition theorem

Abstract

Recently, we have introduced and modified two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product (VRSP), is based on modifications of the well-known Cartesian product, and is closely related to the synchronised product due to Wohrle and Thomas. Here, we recall the definition of the VRSP and the two modified graph-decompositions and introduce three new graph-decomposition theorems. The first new theorem decomposes a graph with respect to the semicomplete bipartite subgraphs of the graph. For the second new theorem, we introduce a matrix graph, which is used to decompose a graph in a manner similar to the decomposition of graphs using the Cartesian product. In the third new theorem, we combine these two types of decomposition. Ultimately, the goal of these graph-decomposition theorems is to come to a prime-graph decomposition.