The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs

Abstract

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs.

The (theta, wheel)-free graphs Part II: Structure theorem

“…Note that for a non-leaf node G i of T the corresponding clique cutset K i is also a clique cutset of G. The following lemmas proved in [7] will also be needed. [7]) If G is a wheel-free graph that contains a diamond, then G has a clique cutset.…”

“…In this series of papers we study the class of (theta, wheel)-free graphs, that we denote by C throughout the paper. This project is motivated and explained in more detail in Part I of the series [7], where two subclasses of C are studied. In Part II of the series [20], we prove a decomposition theorem for graphs in C that uses clique cutsets and 2-joins, and use it to obtain a polynomial time recognition algorithm for the class.…”

The structure of (theta, pyramid, 1‐wheel, 3‐wheel)‐free graphs

“…Given a graph G, its line graph L G ( ) is a graph such that each vertex of L G ( ) represents an edge in G and two vertices of L G ( ) are adjacent if and only if their corresponding edges share a common endpoint in G. A graph is chordless if all of its cycles are chordless. Theorem 1.4 (Diot et al [6]) If G is (theta, pyramid, wheel)-free, then G is the line graph of a triangle-free chordless graph or it admits a clique cutset.…”

“…In Lemma 2.1, we give an n ( ) 6 -time algorithm that decides whether a graph contains a theta, a pyramid, or a 1-wheel. In Lemma 2.2, we give an n ( ) 6 -time algorithm that decides whether a graph contains a 3-wheel. Together these two algorithms give our first recognition algorithm for .…”

“…As already observed, the only Truemper configurations that graphs in may contain are prisms and alternating wheels. Graphs that may contain only prisms (and no other Truemper configuration) are studied in where the following decomposition theorem is obtained. Given a graph , its line graph is a graph such that each vertex of represents an edge in and two vertices of are adjacent if and only if their corresponding edges share a common endpoint in .…”