Related papers: Quasi-kernels in split graphs

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel). Not all directed graphs contain kernels, and computing a kernel or deciding…

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can…

A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e. the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut…

The competition graph of a directed acyclic graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is…

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general…

It is well known that determining if a digraph has a kernel is an NP-complete problem. However, Topp proved that when subdividing every arc of a digraph we obtain a digraph with a kernel. In this paper we define the kernel subdivision…

Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of…

Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least…

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of…

Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $\delta^{0}(D) := \min \{\delta^{+}(D), \delta^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of…

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we…

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…

Most graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of…

In 2001, K\'arolyi, Pach and T\'oth introduced a family of point sets to solve an Erd\H{o}s-Szekeres type problem; which have been used to solve several other Ed\H{o}s-Szekeres type problems. In this paper we refer to these sets as nested…

A graph $G$ with four or more vertices is called bicritical if the removal of any pair of distinct vertices of $G$ results in a graph with a perfect matching. A bicritical graph is minimal if the deletion of each edge results in a…

Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if…

It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further…

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness,…