A decomposition theorem for {ISK4,wheel}-free trigraphs

TL;DR

This paper establishes a structural decomposition theorem for a class of trigraphs that exclude certain complex subgraphs, extending previous results from graphs to trigraphs with undetermined adjacencies.

Contribution

It generalizes a known decomposition theorem for ISK4-free graphs to the broader class of {ISK4,wheel}-free trigraphs, incorporating undetermined adjacencies.

Findings

Proves a new decomposition theorem for {ISK4,wheel}-free trigraphs.

Extends structural understanding from graphs to trigraphs.

Builds on previous work by Lévéque, Maffray, and Trotignon.

Abstract

An ISK4 in a graph G is an induced subgraph of G that is isomorphic to a subdivision of K4 (the complete graph on four vertices). A wheel is a graph that consists of a chordless cycle, together with a vertex that has at least three neighbors in the cycle. A graph is {ISK4,wheel}-free if it has no ISK4 and does not contain a wheel as an induced subgraph. A "trigraph" is a generalization of a graph in which some pairs of vertices have "undetermined" adjacency. We prove a decomposition theorem for {ISK4,wheel}-free trigraphs. Our proof closely follows the proof of a decomposition theorem for ISK4-free graphs due to L\'ev\^eque, Maffray, and Trotignon (On graphs with no induced subdivision of K4. J. Combin. Theory Ser. B, 102(4):924-947, 2012).