Related papers: B1-EPG representations using block-cutpoint trees
Let $\lpt(G)$ be the minimum cardinality of a set of vertices that intersects all longest paths in a graph $G$. Let $\omega(G)$ be the size of a maximum clique in $G$, and $\tw(G)$ be the treewidth of $G$. We prove that $ \lpt(G) \leq…
In p-median location interdiction the aim is to find a subset of edges in a graph, such that the objective value of the p-median problem in the same graph without the selected edges is as large as possible. We prove that this problem is…
A directed graph $G=(V,E)$ is {\it strongly pseudo transitive} if there is a partition $\{A,E-A\}$ of $E$ so that graphs $G_1=(V,A)$ and $G_2=(V,E-A)$ are transitive, and additionally, if $ab\in A$ and $bc\in E $ implies that $ac\in E$. A…
A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na…
Assume that we embed the path $P_n$ as a subgraph of a $2$-dimensional grid, namely, $P_k \times P_l$. Given such an embedding, we consider the ordered set of subpaths $L_1, L_2, \ldots , L_m$ which are maximal straight segments in the…
Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the…
Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181]…
The edge intersection graph of a family of paths in host tree is called an $EPT$ graph. When the host tree has maximum degree $h$, we say that $G$ belongs to the class $[h,2,2]$. If, in addition, the family of paths satisfies the Helly…
We introduce the concept of Most, and Least, Compact Spanning Trees - denoted respectively by $T^*(G)$ and $T^\#(G)$ - of a simple, connected, undirected and unweighted graph $G(V, E, W)$. For a spanning tree $T(G) \in \mathcal{T}(G)$ to be…
The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an…
Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by…
This paper is devoted to a study of single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of a set of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the…
Representation learning on graphs has been gaining attention due to its wide applicability in predicting missing links, and classifying and recommending nodes. Most embedding methods aim to preserve certain properties of the original graph…
Path graphs are intersection graphs of paths in a tree.~In this paper we give a "6\ good characterization" of path graphs, namely, we prove that path graph membership is in $NP\cap CoNP$ without resorting to existing polynomial time…
A graph $G=(V,E)$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ corresponds to a vertex $u \in V$ and there…
Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a,…
A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…
Reconstruction of evolutionary relationships between species is an important topic in the field of computational biology. Pairwise compatibility graphs (PCGs) are used to model such relationships. A graph is a PCG if its edges can be…
Given a finite group $G$ and its representation $\rho$, the corresponding McKay graph is a graph $\Gamma(G,\rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $\pi,\tau$ of…
A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles…