Related papers: Clique cycle-transversals in distance-hereditary g…
We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…
Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph $G$. The nodes of the reduced clique graph are the maximal cliques of $G$, and two nodes are joined by an edge if and only if they form a non-disjoint…
A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in…
Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every…
Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each…
The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique-width if and only if it does not contain all quasi-threshold graphs or their complements. The proof…
We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree…
We generalize the tree-confluent graphs to a broader class of graphs called Delta-confluent graphs. This class of graphs and distance-hereditary graphs, a well-known class of graphs, coincide. Some results about the visualization of…
A cycle double cover (CDC) of an undirected graph is a collection of the graph's cycles such that every edge of the graph belongs to exactly two cycles. We describe a constructive method for generating all the cubic graphs that have a 6-CDC…
Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set…
An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t…
Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If…
We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the…
The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we…
In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…
Arboreal networks are a generalization of rooted trees, defined by keeping the tree-like structure, but dropping the requirement for a single root. Just as the class of cographs is precisely the class of undirected graphs that can be…
The most elusive problem around the class of circular-arc graphs is identifying all minimal graphs that are not in this class. The main obstacle is the lack of a systematic way of enumerating these minimal graphs. McConnell [FOCS 2001]…
In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it…
An orientation of a graph is semi-transitive if it contains no directed cycles and has no shortcuts. An undirected graph is semi-transitive if it can be oriented in a semi-transitive manner. The class of semi-transitive graphs includes…
In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between…