Edge Intersection Graphs Research Articles

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

On the j-Edge Intersection Graph of Cycle Graph

This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called $j$-edge intersection graph of cycle graph, denoted by $E_{C_{(n,j)}}$. The vertex set of $E_{C_{(n,j)}}$ is the set of spanning subgraphs of cycle graph with $j$ edges where $n \geq 3$ and $j$ is a nonnegative integer such that $1 \leq j \leq n$. Moreover, two distinct vertices are adjacent if they have exactly one edge in common. $E_{C_{(n,j)}}$ is considered as a simple graph. Furthermore, $E_{C_{(n,j)}}$ is characterized by the value of $j$ that is when $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$ and $2 \leq j \leq \lceil \frac{n}{2} \rceil$. When $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$, the new graph only produced an empty graph. Hence, the proponents only considered the value when $2 \leq j \leq \lceil \frac{n}{2} \rceil$ in determining the order and size of $E_{C_{(n,j)}}$. Moreover, this paper discusses necessary and sufficient conditions where the $j$-edge intersection graph of $C_n$ is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of $E_{C_{(n,j)}}$ when $j=2$.

On cycle graphs

The Cycle Graph [Formula: see text] of a graph [Formula: see text] is the edge intersection graph of all induced cycles of [Formula: see text]. In this paper, necessary and sufficient condition on [Formula: see text] such that [Formula: see text] is a connected graph, a tree, a bipartite graph is obtained. Non-existence of forbidden subgraph characterization for cycle graphs is also discussed. It is proved that the girth of cycle graph is three. The relationship between radius of [Formula: see text] and its [Formula: see text] is studied. Similar results on diameter and domination number of [Formula: see text] and its cycle graph are also obtained.

Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

In a representation of a graph $G$ as an edge intersection graph of paths on a grid (EPG) every vertex of $G$ is represented by a path on a grid and two paths share a grid edge iff the corresponding vertices are adjacent. In a monotonic EPG representation every path on the grid is ascending in both rows and columns. In a (monotonic) $B_k$-EPG representation every path on the grid has at most $k$ bends. The (monotonic) bend number $b(G)$ ($b^m(G)$) of a graph $G$ is the smallest natural number $k$ for which there exists a (monotonic) $B_k$-EPG representation of $G$. In this paper we deal with the monotonic bend number of outerplanar graphs and show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$ in terms of forbidden induced subgraphs. As a byproduct we obtain low-degree polynomial time algorithms to construct (monotonic) EPG representations with the smallest possible number of bends for maximal outerplanar graphs and cacti.

CPG graphs: Some structural and hardness results

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 in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k≥0, the graph is said to be Bk-CPG.We first show that, for any k≥0, the class of Bk-CPG graphs is strictly contained in the class of Bk+1-CPG graphs even within the class of planar graphs, thus implying that there exists no k≥0 such that every planar CPG graph is Bk-CPG. The main result of the paper is that recognizing CPG graphs and Bk-CPG graphs with k≥1 is NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k≥3. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP-hard for B0-CPG graphs. Finally, we consider the related classes Bk-EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B0-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B1-EPG graphs.

On independent set in [formula omitted]-EPG graphs

In this paper we consider the Maximum Independent Set problem (MIS) on B1-EPG graphs, that is the one-bend (B1) Edge intersection graphs of Paths on a Grid (EPG graphs). The graph class EPG was introduced in Golumbic et al. (2019) as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class Bk-EPG denotes EPG-graphs where every path has at most k bends. The study of MIS on B1-EPG graphs has been initiated in Epstein et al. (2013) where authors prove that MIS is NP-complete on B1-EPG graphs, and provide a polynomial 4-approximation. In this article we study the approximability and the fixed parameter tractability of MIS on B1-EPG. We show that the class of k≥4 subdivided graphs is a subclass of B1-EPG, even if there is only one shape of path and if each path has its vertical part or its horizontal part of length at most 1. This implies that there is no PTAS for MIS (and several other classical problems) on these particular B1-EPG graphs. On the positive side, we show that if the length of the horizontal part of every path is bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on B1-EPG restricted to only three shapes of path, and W[1]-hard on B2-EPG. The status for general B1-EPG (with the four shapes) is left open.

Hardness and approximation for L-EPG and [formula omitted]-EPG graphs

The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths and its subclass of ⌞-shaped paths, namely, the classes known as B1-EPG and ⌞-EPG graphs, respectively.We show that fully-subdivided graphs are ⌞-EPG graphs, and later use this result in order to show that the following problems are APX-hard on ⌞-EPG graphs: Minimum Vertex Cover, Maximum Independent Set, and Maximum Weighted Independent Set, and also that Minimum Dominating Set is NP-complete on ⌞-EPG graphs. We also observe that Minimum Coloring is NP-complete already on ⌞-EPG, which follows from a proof for B1-EPG in Epstein et al. (2013). Finally, we provide efficient constant-factor-approximation algorithms for each of these problems on B1-EPG graphs.

On the Iterated Edge-Biclique Operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The edge-biclique graph of G, KBe(G), is the edge-intersection graph of the bicliques of G. A graph G diverges (resp. converges or is periodic) under an operator H whenever limk→∞ |V (Hk(G))| = ∞ (resp. limk→∞Hk(G) = Hm(G) for some m or Hk(G) = Hk+s(G) for some k and s ≥ 2). The kth-iterated edge-biclique graph of G, KBek(G), is the graph obtained by applying the edge-biclique operator k successive times to G. In this paper we study the iterated edge-biclique operator KBe. In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator KBe and we propose some conjectures on the subject.

Edge-intersection graphs of boundary-generated paths in a grid

Edge-intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent in the graph if and only if the corresponding paths share at least one edge of the grid. For two boundary points p and q on two adjacent boundaries of a rectangular grid G, we call the unique single-bend path connecting p and q in G using no other boundary point of G as the path generated by (p,q). A path in G is called boundary-generated, if it is generated by some pair of points on two adjacent boundaries of G. In this article, we study the edge-intersection graphs of boundary-generated paths on a grid or ∂EPG graphs. The motivation for studying these graphs comes from problems in the context of circuit layout.We show that ∂EPG graphs can be covered by two collections of vertex-disjoint co-bipartite chain graphs. This leads us to a linear-time testable characterization of ∂EPG trees and also an almost tight upper bound on the equivalence covering number of general ∂EPG graphs. We also study the cases of two-sided ∂EPG and three-sided ∂EPG graphs, which are respectively, the subclasses of ∂EPG graphs obtained when all the boundary-vertex pairs which generate the paths are restricted to lie on at most two or three boundaries of the grid. For the former case, we give a complete characterization.

Helly EPT graphs on bounded degree trees: Characterization and recognition

The edge-intersection graph of a family of paths on a host tree is called an EPT graph. When the tree has maximum degree h, we say that the graph is [h,2,2]. If, in addition, the family of paths satisfies the Helly property, then the graph is Helly [h,2,2]. In this paper, we present a family of EPT graphs called gates which are forbidden induced subgraphs for [h,2,2] graphs. Using these we characterize by forbidden induced subgraphs the Helly [h,2,2] graphs. As a byproduct we prove that in getting a Helly EPT-representation, it is not necessary to increase the maximum degree of the host tree. In addition, we give an efficient algorithm to recognize Helly [h,2,2] graphs based on their decomposition by maximal clique separators.

On edge intersection graphs of paths with 2 bends

An EPG-representation of a graph G is a collection of paths in the 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 this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses.We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes.

Clique coloring [formula omitted]-EPG graphs

We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] 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 rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection graph of paths on a grid, i.e., one can associate to each vertex of the graph a nontrivial path on a grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is B4-EPG, by embedding the circle into a rectangle of the grid. In this paper we prove that every circular-arc graph is B3-EPG, but if we restrict ourselves to rectangular representations there exist some graphs that require paths with four bends. We also show that normal circular-arc graphs admit rectangular representations with at most two bends per path. Moreover, we characterize graphs admitting a rectangular representation with at most one bend per path by forbidden induced subgraphs, and we show that they are a subclass of normal Helly circular-arc graphs.

Graphs of edge-intersecting and non-splitting paths

The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and Non-Splitting Paths in a Tree (ENPT) [5]. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph class by allowing the host graph to be an arbitrary graph. These are the graphs of Edge-Intersecting and Non-Splitting Paths ENP. Although the Edge Intersection Graphs of Paths in an arbitrary graph includes all graphs, we show that this is not the case for ENP. We also show that the class ENP coincides with the family of graphs of Edge-Intersecting and Non-Splitting Paths in a Grid (ENPG). Following similar studies for EPG graph class, we study the implications of restricting the number of bends of the individual paths in the grid. We show that such a restriction restricts the graph class. Specifically, by restricting the number of bends one gets an infinite sequence of classes such that every class is properly included in the next one.

Strong cliques and equistability of EPT graphs

In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.

Graphs of edge-intersecting non-splitting paths in a tree: Representations of holes—Part I

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T, and Ept(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices are the vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path).In this work, motivated by an application in all-optical networks, we define the graph Enpt(P) of edge-intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in P, and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=Enpt(P), and we say that 〈T,P〉 is a representation of G. We first show that cycles, trees and complete graphs are ENPT graphs.Our work follows the lines of Golumbic and Jamison’s research (Golumbic and Jamison, 1985) in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). It turns out that ENPT holes have a more complex structure than EPT holes. In our analysis, we assume that the EPT graph corresponding to a representation of an ENPT hole is given. We also introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In this Part I, using the results of Golumbic and Jamison as building blocks, we characterize (a) EPT, ENPT pairs that satisfy (P1)–(P3), and (b) the unique minimal representation of such pairs.

Edge intersection graphs of [formula omitted]-shaped paths in grids

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: ⌞,⌜,⌟,⌝, and we consider zero bend paths (i.e., ∣ and–) to be degenerate ⌞’s. These graphs, called B1-EPG graphs, were first introduced by Golumbic et al. (2009). We consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝}, and {⌞,⌜,⌝}) and we denote the classes by [⌞],[⌞,⌜],[⌞,⌝], and [⌞,⌜,⌝] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩[⌞].

On local edge intersection graphs of paths on bounded degree trees

On local edge intersection graphs of paths on bounded degree trees