Related papers: All-path convexity: Combinatorial and complexity a…

Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…

Without imposing restrictions on a weighted graph's arc lengths, symmetry structures cannot be expected. But, they exist. To find them, the graphs are decomposed into a component that dictates all closed path properties (e.g., shortest and…

For a graph $G=(V,\ E)$ and a nonempty set $S\subseteq V$, the \emph{vertex boundary} of $S$, denoted by $\partial_G(S)$, is defined to be the set of vertices that are not in $S$ but have at least one neighbor in $S$. In this paper, for $G$…

For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$ in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by $\psi_k(G)$.…

Graph convexity spaces have been studied in many contexts. In particular, some studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. It is well known that chordal and Ptolemaic graphs can be…

For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing…

We study convexity of image of a general multidimensional quadratic map. We split the full image into two parts by an appropriate hyperplane such that one part is compact, and formulate a sufficient condition for convexity of the compact…

We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in…

Given an undirected graph G and a set A \subseteq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) \setminus A. An A-path is called an (A,\ell)-path if the length of the…

Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one…

For a graph $G$ with vertex assignment $c:V(G)\to \mathbb{Z}^+$, we define $\sum_{v\in V(H)}c(v)$ for $H$ a connected subgraph of $G$ as a connected subgraph sum of $G$. We study the set $S(G,c)$ of connected subgraph sums and, in…

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the…

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two…

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for…

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the…

In this paper we generalize a specific quantized convexity structure of the generalized state space of a $C^*$-algebra and examine the associated extreme points. We introduce the notion of $P$-$C^*$-convex subsets, where $P$ is any positive…

We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by…

A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…

The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…

Given a connected graph $G=(V(G), E(G))$, the length of a shortest path from a vertex $u$ to a vertex $v$ is denoted by $d(u,v)$. For a proper subset $W$ of $V(G)$, let $m(W)$ be the maximum value of $d(u,v)$ as $u$ ranging over $W$ and $v$…