Unit Incomparability Dimension and Clique Cover Width in Graphs

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 $CCW (G)\le s(G)-1$.