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We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct…
We study the expressivity of sparse maxout networks, where each neuron takes a fixed number of inputs from the previous layer and employs a, possibly multi-argument, maxout activation. This setting captures key characteristics of…
From the paper of the first author it follows that upper and lower bounds for $\gamma$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and…
The stable set polytope of a graph $G$, denoted as STAB($G$), is the convex hull of all the incidence vectors of stable sets of $G$. To describe a linear system which defines STAB($G$) seems to be a difficult task in the general case. In…
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
Given a directed graph $D=(V,A)$ with a set of $d$ specified vertices $S=\{s_1,...,s_d\}\subseteq V$ and a function $f\colon S \to \mathbb{Z}_+$ where $\mathbb{Z}_+$ denotes the set of non-negative integers, we consider the problem which…
We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as…
Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if there exists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a…
The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$,…
We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.
Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the…
A perfect matching in an undirected graph $G=(V,E)$ is a set of vertex disjoint edges from $E$ that include all vertices in $V$. The perfect matching problem is to decide if $G$ has such a matching. Recently Rothvo{\ss} proved the striking…
We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…
We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more specifically, of graph classes with bounded tree-independence number. In [Dallard, Milani\v{c}, and…
What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall…
This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank…
It is known that the extension complexity of the TSP polytope for the complete graph $K_n$ is exponential in $n$ even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets…
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…
Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F form an increasing sequence. Hallam and Sagan showed that the generating…