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A $k$-bend path is a non-self-intersecting polyline in the plane made of at most $k+1$ axis-parallel line segments. B$_k$-VPG is the class of graphs which can be represented as intersection graphs of $k$-bend paths in the same plane. In…
We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at…
A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A $k$-bend path is a path made up of at most $k + 1$ line segments. A $B_k$-VPG representation of a graph is a collection…
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…
A graph $G$ is called $B_k$-VPG, for some constant $k\geq 0$, if it has a string representation on an axis-parallel grid such that each vertex is a path with at most $k$ bends and two vertices are adjacent in $G$ if and only if the…
We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We show similarities and differences from this new class to the well-known class of EPG graphs. A turn of a…
If a graph $G$ can be represented by means of paths on a grid, such that each vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge, then this graph…
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, \Gamma, LE{} and \eeG. A $k$-bend path is a simple path in the plane, whose direction changes $k$ times from horizontal…
In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes:…
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…
A piecewise linear curve in the plane made up of $k+1$ line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a $k$-bend path. Given a graph $G$, a collection of…
In this paper, we prove that every planar graph has a 1-string $B_2$-VPG representation---a string representation using paths in a rectangular grid that contain at most two bends. Furthermore, two paths representing vertices $u,v$ intersect…
Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a…
Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$,…
In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn).
An EPG-representation of a graph $G$ is a collection of paths in a plane square grid, each corresponding to a single vertex of $G$, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In…
A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact…
In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are…
The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid…
Given a graph $G(V, E)$ and a positive integer $k$ ($k \geq 1$), a simple path on $k$ vertices is a sequence of $k$ vertices in which no vertex appears more than once and each consecutive pair of vertices in the sequence are connected by an…