Related papers: Edge Intersection Graphs of Paths on a Triangular …
An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The…
An edge coalition in a graph $G=(V,E)$ consists of two disjoint sets of edges $E_1$ and $E_2$, neither of which is an edge dominating set but whose union $E_1\cup E_2$ is an edge dominating set. An edge coalition partition in a graph $G$ of…
A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius…
An edge-girth-regular graph $egr(v,k,g,\lambda)$, is a $k$-regular graph of order $v$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g$-cycles. An $egr(v,k,g,\lambda)$ is called extremal…
We consider Independent Set and Dominating Set restricted to VPG graphs (or, equivalently, string graphs). We show that they both remain $\mathsf{NP}$-hard on $B_0$-VPG graphs admitting a representation such that each grid-edge belongs to…
A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2.…
Edge-girth-regular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no $3$-regular \emph{egr} graph with girth $7$ such that every…
An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of…
A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E_1,E_2 such that |E_1| = k and…
Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…
An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…
We prove that for every graph $G$, given fixed locations for the vertices of $G$ in $\mathbb{Z}^3$, there is a three-dimensional grid-drawing of $G$ with one bend per edge. The best previous bound was three bends per edge.
Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to…
In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the…
We define the \emph{occurrence graph} $G_p(\pi$) of a pattern $p$ in a permutation $\pi$ as the graph with the occurrences of $p$ in $\pi$ as vertices and edges between the vertices if the occurrences differ by exactly one element. We then…
A graph $G$ is a \emph{max point-tolerance (MPT)} graph if each vertex $v$ of $G$ can be mapped to a \emph{pointed-interval} $(I_v, p_v)$ where $I_v$ is an interval of $\mathbb{R}$ and $p_v \in I_v$ such that $uv$ is an edge of $G$ iff $I_u…
An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval…
It was shown by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in…
We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as…
In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…