Related papers: Obstruction characterization of co-TT graphs
We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called the simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined…
We consider the following fundamental realization problem of directed graphs. Given a sequence $S:={a_1 \choose b_1},\dots,{a_n \choose b_n}$ with $a_i,b_i\in \mathbb{Z}_0^+.$ Does there exist a digraph (no loops and no parallel arcs are…
Bir\'{o}, Hujter, and Tuza introduced the concept of $H$-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph $H$. They naturally generalize many important classes of graphs, e.g., interval graphs and…
The \emph{thinness} of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many…
Jones et al. (2015) introduced the notion of $u$-representable graphs, where $u$ is a word over $\{1, 2\}$ different from $22\cdots2$, as a generalization of word-representable graphs. Kitaev (2016) showed that if $u$ is of length at least…
A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are…
Bir\'{o} et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split…
A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if…
In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called $k$-length interval graphs were considered in which the number of different lengths…
Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every $t\geq 1$, the $t$-basic obstructions: the graphs $K_{t+1}$ and $K_{t,t}$, along with the subdivisions of the $t$-by-$t$…
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]…
Lettericity is a graph parameter responsible for many attractive structural properties. In particular, graphs of bounded lettericity have bounded linear clique-width and they are well-quasi-ordered by induced subgraphs. The latter property…
A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are…
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a…
Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower…
Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of…
The problem of finding dense induced bipartite subgraphs in $H$-free graphs has a long history, and was posed 30 years ago by Erd\H{o}s, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove…
We introduce a variation of interval graphs, called veto interval (VI) graphs. A VI graph is represented by a set of closed intervals, each containing a point called a veto mark. The edge $ab$ is in the graph if the intervals corresponding…
We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called $B_0$-VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time…
The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph,…