On Strict Brambles

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ 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 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 ${\sf sbn}(G)$ 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 ${\sf sbn}(G)$ 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 ${\sf sbn}(G) \leq k$ is an ${\sf NP}$-complete problem.