On Strict Brambles

TL;DR

The paper formalizes the strict bramble number $sbn(G)$ and proves three equivalent viewpoints: a min-max relation with $ltp(G)$ via lexicographic tree products, a width measure via lenient tree decompositions, and an extremal structure via $k$-domino-trees. It establishes a tight equivalence among these perspectives and identifies the minor-obstruction sets for small $k$, notably ${\sf Obs}({\cal G}_2)={\cal Z}$. It also proves that computing $sbn(G)$ is NP-hard (in fact NP-complete), tying the parameter to treewidth through a reduction, and shows that edge-maximal graphs with $sbn\le k$ are precisely the (partial) $k$-domino-trees with tight edge bounds. The results provide a foundational framework linking acyclicity-variants to tree-like decompositions and minor-closed classifications, with potential algorithmic implications for graph width parameters and related combinatorial problems.

Abstract

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.

Figures (7)

• Figure 1: A lenient tree decomposition of the graph $G'$ of \ref{['domino_expl']}.
• Figure 2: A graph $G$ with a set $S$ of three square vertices, such that $G'=G\setminus S$ is a $2$-domino-tree. The neighbor of the red square vertex is a cut-vertex, which violates property \ref{['@kennzeichnet']}. Also, the $K_{2}$ incident to the red square, violates property \ref{['@correspondiente']}. The violet square vertex cannot exist, as otherwise the $K_{3}$ induced by it and its neighbors is separated by $\{i,f\},$ which is contained in the maximal clique induced by $\{f,g,h,i\}$ which contains three minimal separators. This violates property \ref{['@begriffswort']}. The maximal clique induced by the green square vertex and its neighbors contains two minimal separators that do not cover the green square vertex. This violates property \ref{['@precapitalist']}. Consider now the graph $G'.$ If we remove from $G'$ the edge $\{c,d\},$ the graph will no longer be chordal, which violates property \ref{['@unswervingly']}. If we remove the edge $\{k,l\}$, the set $\{j,m\}$ becomes an internal minimal separator of connectivity-degree two, that is contained in the union of the minimal separators $\{j,k\}$ and $\{l,m\},$ which violates property \ref{['@mandamientos']}. If we remove the edge $\{l,o\},$ the set $\{m,n\}$ becomes an external minimal separator of connectivity-degree two, where the union of the vertex sets of the maximal cliques induced by $\{l,m,n\}$ and $\{m,n,o\},$ has size four, which violates \ref{['@presentiment']}. If we remove edge $\{a,b\},$ the external minimal separator $\{c,d\}$ belongs to the external maximal cliques induced by $\{a,c,d\}$ and $\{b,c,d\}$ and the sum of the valiances of these two cliques is two, which violates property \ref{['@bezeichnenden']}.
• Figure 3: The connected component $C,$ and the minimal separators $S$ and $S'$ of \ref{['@manifestations']}.
• Figure 4: A $2$-domino-tree on $8$ vertices, and $16$ edges, which is the maximum possible number of edges a $2$-domino-tree on $8$ vertices may have.
• Figure 5: A $2$-domino-tree on $8$ vertices and $15$ edges.

Theorems & Definitions (43)

• Lemma 1
• proof
• Proposition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5