Finding the maximum number of maximal independent sets in an $n$-vertex graph $G$, $i(G)$, from a restricted class is an extensively studied problem. Let $kK_2$ denote the matching of size $k$, that is a graph with $2k$ vertices and $k$ disjoint edges. A graph with an induced copy of $kK_2$ contains at least $2^k$ maximal independent sets. The other direction was established in a series of papers finally yielding $i(G) \le (n/k)^{2k}$ for a graph $G$ without an induced $(k+1)K_2$. Alekseev proved that $i(G)$ is at most the number of induced matchings of $G$.
This work generalises the aforementioned results to clutters. The right substructures in this setting are minors rather than induced subgraphs. Maximal independent sets of a clutter $\mathcal{H}$ are in one-to-one correspondence to the sets of its blocker, $b(\mathcal{H})$, hence $i(\mathcal{H}) = |b(\mathcal{H})|$. We show that
\[
|b(\mathcal{H})| \le
\sum_{m=0}^{k \cdot f(r)}{|\mathcal{H}| \choose m} {r \choose 2}^m
\] for a $(k+1)K_2$-minor-free clutter $\mathcal{H}$ where $f(r) = (2r-3)2^{r-2}$ and $r$ is the maximum size of a set in $\mathcal{H}$. A key step in the proofs is, similarly to Alekseev's result, showing that $i(\mathcal{H})$ is bounded by the number of a substructure called semi-matching, and then proving a dependence between the number of semi-matchings and the number of minor matchings. Note that similarly to graphs, a clutter containing a $kK_2$ minor has at least $2^k$ maximal independent sets.
From a computational perspective, a polynomial number of independent sets is particularly interesting. Our results lead to polynomial algorithms for restricted instances of many problems including Set Cover and k-SAT.
Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can becomputed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introducea version of Erdos-Posa property for matching minors and find a direct link between this property and planarity. From…
A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. Our result builds on a a tight relationship between structural digraph theory and matching theory and allows us to deduce a Flat Wall Theorem…
\noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $\#\mathsf{P}$-hard. In 1913 Pólya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a \{matching minor}. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in…
Let $c(H)$ be the smallest value for which $e(G)/|G|\geq c(H)$ implies $H$ is a minor of $G$. We show a new upper bound on $c(H)$, which improves previous bounds for graphs with a vertex partition where some pairs of parts have many more edges than others -- for instance a complete bipartite graph with a small number of edges placed inside one class. We also show a tight matching lower bound for almost all such graphs. We apply these results to show $c(K_{ft/\log t,t}) = (0.638\dotsc+o_{f}(1))t\sqrt{f}$, for $f = o(\log t) = ω(1)$.
Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. We say that a graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable. In this paper we consider the next step and prove that every graph with no $\mathcal{K}_8^{-4}$ min…
Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For fixed $s\ge 2$ and sufficiently large $n$, the $F_s$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)],and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no $H$ as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (…
Center for Combinatorics and LPMC, Nankai University（南开大学组合数学与LPMC中心）；Department of Mathematics, Paderborn University（帕德博恩大学数学系）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
