Related papers: Counting matchings with k unmatched vertices in pl…
Enumerating matchings is a classical problem in the field of enumeration algorithms. There are polynomial-delay enumeration algorithms for several settings, such as enumerating perfect matchings, maximal matchings, and (weighted) matchings…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of…
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After…
Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…
Given a symmetric D*D matrix M over {0,1,*}, a list M-partition of a graph G is a partition of G's vertices into D parts associated with the rows of M. The part of each vertex is chosen from a given list so that no edge of G maps to a 0 in…
It is well-known that every $n$-vertex planar graph with minimum degree 3 has a matching of size at least $\frac{n}{3}$. But all proofs of this use the Tutte-Berge-formula for the size of a maximum matching. Hence these proofs are not…
A matching-cut of a graph is an edge cut that is a matching. The problem Matching-Cut is that of recognizing graphs with a matching-cut and is NP-complete, even if the graph belongs to one of a number of classes. We initiate the study of…
We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an…
We study the $P_3$-convexity, the path convexity generated by all three-vertex paths, and focus on the problem of counting the $P_3$-convex vertex sets of a graph $G$, denoted by $\noc(G)$. First, we settle the associated extremal question:…
In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least…
$ $We study the $d$-Uniform Hypergraph Matching ($d$-UHM) problem: given an $n$-vertex hypergraph $G$ where every hyperedge is of size $d$, find a maximum cardinality set of disjoint hyperedges. For $d\geq3$, the problem of finding the…
Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our goal is to construct a matching in $G$ that is popular. A matching $M$ is popular if $M$ does not lose a…
We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any…
General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…
Let $H$ be a $k$-graph on $n$ vertices, with minimum codegree at least $n/k + cn$ for some fixed $c > 0$. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in $H$ or a certificate that none exists.…
Counting maximum matchings in a graph is of great interest in statistical mechanics, solid-state chemistry, theoretical computer science, mathematics, among other disciplines. However, it is a challengeable problem to explicitly determine…
The Disjoint Paths Problem asks, given a graph $G$ and a set of pairs of terminals $(s_{1},t_{1}),\ldots,(s_{k},t_{k})$, whether there is a collection of $k$ pairwise vertex-disjoint paths linking $s_{i}$ and $t_{i}$, for $i=1,\ldots,k.$ In…
introduce {\sc Planar Disjoint Paths Completion}, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph $G,$ $k$…
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…