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

The 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) = ∩di=1(Gi), then we write G = ∩di=1Gi and call each Gi, i = 1, 2, ..., d a factor of G. The case where G1 is chordal, and for i = 2, 3..., d each factor Gi has a “small” ccw(Gi), is well studied due to applications. Here we show a negative result. Specifically, let Ĝ(k, n) be the graph obtained by joining a set of k apex vertices of degree n to all vertices of an n×n grid, and then adding some possible edges among these k vertices. We prove that if Ĝ(k, n) = ∩di=1Gi, with G1 being chordal, then, max2≤i≤d{ccw(Gi)} = Ω(n 1 d−1 ). Furthermore, for d = 2, we construct a chordal graph G1 and a graph G2 with ccw(G2) ≤ n 2 + k so that Ĝ(k, n) = G1 ∩ G2. Finally, let Ĝ be the clique sum graph of Ĝ(ki, ni), where for i = 1, 2, ...t, the underlying grid is ni × ni and the sum is taken at apex vertices. Then, we show Ĝ = G1 ∩G2, where, G1 is chordal and ccw(G2) ≤ ∑ t i=1 (ni+ki). The implications and applications of the results are discussed, including addressing a recent question of David Wood.