Unit Incomparability Dimension and Clique Cover Width in Graphs

TL;DR

This paper introduces the concepts of clique cover width and unit incomparability dimension in graphs, establishing bounds and decompositions that relate these parameters to graph structures like stars and Ramsey theory.

Contribution

It presents a decomposition theorem linking unit incomparability dimension to clique cover width and provides bounds based on graph star structures and Ramsey theory.

Findings

Udim(G) CW(G) for any graph G

Decomposition of G into unit incomparability graphs with specific properties

Bounds on clique cover width using star leaves and Ramsey theory

Abstract

For a clique cover $C$ in the undirected graph $G$, the {\it clique cover graph} of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The {\it clique cover width} of $G$, denoted by $CCW(G)$, is the minimum value of the bandwidth of all clique cover graphs in $G$. Any $G$ with $CCW(G)=1$ is known to be an incomparability graph, and hence is called, a {\it unit incomparability graph}. We introduced the {\it unit incomparability dimension of $G$}, denoted by$Udim(G)$, to be the smallest integer $d$ so that there are unit incomparability graphs $H_i$ with $V(H_i)=V(G), i=1,2,...,d$, so that $E(G)=\cap_{i=1}^d E(G_i)$. We prove a decomposition theorem establishing the inequality $Udim(G)\le CCW(G)$. Specifically, given any $G$, there are unit incomparability graphs $H_1,H_2,...,H_{CC(W)}$ with $V(H_i)=V(G)$ so that and $E(G)=\cap_{i=1}^{CCW} E(H_i)$. In addition, $H_i$ is co-bipartite, for $i=1,2,...,CCW(G)-1$. Furthermore, we observe that $CCW(G)\ge s(G)/2-1$, where $s(G)$ is the number of leaves in a largest induced star of $G$ , and use Ramsey Theory to give an upper bound on $s(G)$, when $G$ is represented as an intersection graph using our decomposition theorem. Finally, when $G$ is an incomparability graph we prove that $CCW (G)\le s(G)-1$.