Related papers: Separation axiom $S_3$ for geodesic convexity in g…

Convexity is a notion that has been defined for subsets of $\RR^n$ and for subsets of general graphs. A convex cut of a graph $G=(V, E)$ is a $2$-partition $V_1 \dot{\cup} V_2=V$ such that both $V_1$ and $V_2$ are convex, \ie shortest paths…

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the…

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$…

A graph $G=(V,E)$ is a {\it unipolar graph} if there exits a partition $V=V_1 \cup V_2$ such that, $V_1$ is a clique and $V_2$ induces the disjoint union of cliques. The complement-closed class of {\it generalized split graphs} are those…

\noindent A perfect dominating set in a graph $G=(V,E)$ is a subset $S \subseteq V$ such that each vertex in $V \setminus S$ has exactly one neighbor in $S$. A perfect coalition in $G$ consists of two disjoint sets of vertices $V_i$ and…

We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean…

In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to…

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius $r$ is called an $r$-ASC graph. The $r$-ASC index $\theta_r(G)$ of a graph $G$ is the minimum number of vertices…

A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such…

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex…

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm…

In this paper, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an…

A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of…

For a bridgeless cubic graph $G$, $m_3(G)$ is the ratio of the maximum number of edges of $G$ covered by the union of $3$ perfect matchings to $|E(G)|$. We prove that for any $r\in [4/5, 1)$, there exist infinitely many cubic graphs $G$…

Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d \colon V \times V \to [0, \infty[$ and disjoint proximinal subsets of the semimetric space $(V, d)$ such that $V = A \cup B$, and…

The inclusion ideal graph $\mathcal{I}n(S)$ of a semigroup $S$ is an undirected simple graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if either $I \subset J$ or $J…

A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic…

A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all…