Bounds for the Clique Cover Width of Factors of the Apex Graph of the Planar Grid

The {\it clique cover width} of $G$, denoted by $ccw(G)$, is the minimum value of the bandwidth of all graphs that are obtained by contracting the cliques in a clique cover of $G$ into a single vertex. For $i=1,2,...,d,$ let $Gi$ be a graph with $V(Gi)=V$, and let $G$ be a graph with $V(G)=V$ and $E(G)=\cap{i=1}^d(Gi)$, then we write $G=\cap{i=1}^dGi$ and call each $Gi,i=1,2,...,d$ a factor of $G$. We are interested in the case where $G1$ is chordal, and $ccw(Gi),i=2,3...,d$ for each factor $Gi$ is "small". Here we show a negative result. Specifically, let ${\hat G}(k,n)$ be the graph obtained by joining a set of $k$ apex vertices of degree $n^2$ to all vertices of an $n\times n$ grid, and then adding some possible edges among these $k$ vertices. We prove that if ${\hat G}(k,n)=\cap{i=1}^dGi$, with $G1$ being chordal, then, $max{2\le i\le d}{ccw(Gi)}\ge {n^{1\over d-1}\over 2.{(2c)}^{1\over {d-1}}}$, where $c$ is a constant. Furthermore, for $d=2$, we construct a chordal graph $G1$ and a graph $G2$ with $ccw(G2)\le {n\over 2}+k$ so that ${\hat G}(k,n)=G1\cap G2$. Finally, let ${\hat G}$ be the clique sum graph of ${\hat G}(ki, ni), i=1,2,...t$, where the underlying grid is $ni\times ni$ and the sum is taken at apex vertices. Then, we show ${\hat G}=G1\cap G2$, where, $G1$ is chordal and $ccw(G2)\le \sum{i=1}^t(ni+k_i)$. The implications and applications of the results are discussed, including addressing a recent question of David Wood.