Excluded-Minor Characterization of Apex-Outerplanar Graphs

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: (Formula presented.) and (Formula presented.). The class of graphs that contain a vertex whose removal leaves an outerplanar graph is also minor-closed. We provide the complete list of 57 excluded minors for this class

Bandwidth of trees of diameter at most 4

For a graph G, let γ:V(G)→1,⋯,|V(G)| be a one-to-one function. The bandwidth of γ is the maximum of |γ(u)-γ(v)| over uv∈E(G). The bandwidth of G, denoted b(G), is the minimum bandwidth over all embeddings γ, b(G)=min γmax|γ(u)-γ(v) |:uv∈E(G). In this paper, we show that the bandwidth computation problem for trees of diameter at most 4 can be solved in polynomial time. This naturally complements the result computing the bandwidth for 2-caterpillars. © 2012 Elsevier B.V. All rights reserved

On 3-connected graphs of path-width at most three

Tree-width and path-width are two important graph parameters introduced by Robertson and Seymour in their famous Graph Minors project. For a fixed positive integer k, the classes of graphs of tree-width at most k, and path-width at most k, are both minor-closed. However, their complete characterizations in terms of excluded minors are known only for k ≤ 3 for tree-width, and for k ≤ 2 for path-width. It is known that the number of excluded minors for the class of graphs of path-width ≤ k is 2 if k = 1; 110 if k = 2; and ≥ 122 million if k = 3. Baŕat et. al. [Studia Sci. Math. Hungar., 49 (2012), pp. 211-222] showed that the class of graphs of path-width ≤ 2, restricted to its 2-connected members, can be characterized by only three excluded minors, and asked whether a similar result may be obtained for 3-connected graphs of path-width ≤ 3. We answer this question in the affirmative by characterizing this class by five excluded minors. ©2013 Society for Industrial and Applied Mathematics

Vertex-Bipartition Method for Colouring Minor-Closed Classes of Graphs

Thomas conjectured that there is an absolute constant c such that for every proper minor-closed class of graphs, there is a polynomial-time algorithm that can colour every G ∈ C with at most χ (G) + c colours. We introduce a parameter , called the degenerate value of C, which is defined to be the smallest r such that every G ∈ can be vertex-bipartitioned into a part of bounded tree-width (the bound depending only on ), and a part that is r-degenerate. Although the existence of one global bound for the degenerate values of all proper minor-closed classes would imply Thomas\u27s conjecture, we prove that the values can be made arbitrarily large. The problem lies in the clique sum operation. As our main result, we show that excluding a planar graph with a fixed number of apex vertices gives rise to a minor-closed class with small degenerate value. As corollaries, we obtain that (i) the degenerate value of every class of graphs of bounded local tree-width is at most 6, and (ii) the degenerate value of the class of Kn-minor-free graphs is at most n + 1. These results give rise to P-time approximation algorithms for colouring any graph in these classes within an error of at most 7 and n + 2 of its chromatic number, respectively. Copyright © 2010 Cambridge University Press