Related papers: Path Graphs, Clique Trees, and Flowers
We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers to the negative a question of Carbonero, Hompe, Moore,…
We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of…
In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [Deniz et al., 2018]. A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid…
Let $D$ be a multidigraph. We study the simplicial complex $\mathrm{Dlf}(D)$, whose vertices are the directed edges of $D$ and whose faces correspond to directed linear forests, that is, vertex-disjoint unions of directed paths. We also…
Extending the well-known star-comb lemma for infinite graphs, we characterise the graphs that do not contain an infinite comb or an infinite star, respectively, attached to a given set of vertices. We offer several characterisations: in…
We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called $\alpha_i$-metric ($i\in \mathcal{N}$) if it…
For a family $\mathcal{H}$ of graphs, a graph $G$ is said to be {\it $\mathcal{H}$-free} if $G$ contains no member of $\mathcal{H}$ as an induced subgraph. We let $\tilde{\mathcal{G}}_{3}(\mathcal{H})$ denote the family of connected…
Directed acyclic graphs whose nodes are all the divisors of a positive integer $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered. Fourteen graph invariants such as order, size, and the number of paths are investigated for two…
A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of…
Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median…
A \emph{wheel} is a graph formed by a chordless cycle $C$ and a vertex $u$ not in $C$ that has at least three neighbors in $C$. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a…
We introduce the \emph{ID-index} of a finite simple connected graph. For a graph $G=(V,\ E)$ with diameter $d$, we let $f:V\longrightarrow \mathbb{R}$ assign \emph{ranks} to the vertices, then under $f$, each vertex $v$ gets a…
A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an…
In this paper we study locally chordal graphs, i.e. graphs where every small-radius ball is chordal. We prove four characterizations of locally chordal graphs. Two are counterparts of the classic descriptions of chordal graphs via induced…
Given a drawing of a graph, its \emph{visual complexity} is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in…
We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by…
A graph is circle if there is a family of chords in a circle such that two vertices are adjacent if the corresponding chords cross each other. There are diverse characterizations of circle graphs, many of them using the notions of local…
In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial…
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to…