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

TL;DR

This paper investigates bounds on the clique cover width of factors of the apex graph of a planar grid, providing negative results for certain decompositions and constructions for specific cases, with implications for graph theory questions.

Contribution

It establishes lower bounds on clique cover width for apex graph factors with a chordal component and constructs explicit decompositions for particular cases.

Findings

Lower bounds on ccw for apex graph factors with chordal components

Explicit constructions for d=2 case

Decomposition of apex graphs into chordal and bounded ccw factors

Abstract

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 $G_i$ be a graph with $V(G_i)=V$, and let $G$ be a graph with $V(G)=V$ and $E(G)=\cap_{i=1}^d(G_i)$, then we write $G=\cap_{i=1}^dG_i$ and call each $G_i,i=1,2,...,d$ a factor of $G$. We are interested in the case where $G_1$ is chordal, and $ccw(G_i),i=2,3...,d$ for each factor $G_i$ 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 ${\hat G}(k,n)=\cap_{i=1}^dG_i$, with $G_1$ being chordal, then, $max_{2\le i\le d}\{ccw(G_i)\}\ge {n^{1\over d-1}\over 2.{(2c)}^{1\over {d-1}}}$, where $c$ is a constant. Furthermore, for $d=2$, we construct a chordal graph $G_1$ and a graph $G_2$ with $ccw(G_2)\le {n\over 2}+k$ so that ${\hat G}(k,n)=G_1\cap G_2$. Finally, let ${\hat G}$ be the clique sum graph of ${\hat G}(k_i, n_i), i=1,2,...t$, where the underlying grid is $n_i\times n_i$ and the sum is taken at apex vertices. Then, we show ${\hat G}=G_1\cap G_2$, where, $G_1$ is chordal and $ccw(G_2)\le \sum_{i=1}^t(n_i+k_i)$. The implications and applications of the results are discussed, including addressing a recent question of David Wood.