On Strict Brambles [article]

Emmanouil Lardas and Evangelos Protopapas and Dimitrios M. Thilikos and Dimitris Zoros
2022 arXiv   pre-print
A strict bramble of a graph G is a collection of pairwise-intersecting connected subgraphs of G. The order of a strict bramble B is the minimum size of a set of vertices intersecting all sets of B. The strict bramble number of G, denoted by sbn(G), is the maximum order of a strict bramble in G. The strict bramble number of G can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order
more » ... one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that sbn(G) is equal to the minimum k for which G is a minor of the lexicographic product of a tree and a clique on k vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that sbn(G) is equal to the minimum k for which there exists a lenient tree decomposition of G of width at most k. The third characterization is in terms of extremal graphs. For this, we define, for each k, the concept of a k-domino-tree and we prove that every edge-maximal graph of strict bramble number at most k is a k-domino-tree. We also identify three graphs that constitute the minor-obstruction set of the class of graphs with strict bramble number at most two. We complete our results by proving that, given some G and k, deciding whether sbn(G) ≤ k is an NP-complete problem.
arXiv:2201.05783v1 fatcat:vhqb3gfn2ves7j4c33unlq3q6q