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.
Publication Source (Journal or Book title)
SIAM Journal on Discrete Mathematics
Ding, G., & Dziobiak, S. (2013). On 3-connected graphs of path-width at most three. SIAM Journal on Discrete Mathematics, 27 (3), 1514-1526. https://doi.org/10.1137/100800452