Edge Intersection Graphs Research Articles

On the bend-number of planar and outerplanar graphs

The bend-numberb(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bounds for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.

Edge-intersection graphs of grid paths: The bend-number

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 least one grid-edge. The bend-number is the minimum k such that grid-paths with at most k bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every k there is a graph with bend-number k. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of Km,n and determine it exactly for some pairs of m and n. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.

Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞,⌜,⌟,⌝). These graphs, called B1- EPG graphs, were first introduced by Golumbic et al (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid, Networks 54:3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split∩[⌞]. We also consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝},{⌞,⌜,⌝} – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]).

Single bend paths on a grid have strong helly number 4: errata atque emendationes ad “edge intersection graphs of single bend paths on a grid”

AbstractIn this note, we prove that a collection of single bend paths on a grid, the so‐called B1 representations, have strong Helly number 4. This corrects a false claim from Ref. [Golumbic et al., Networks, 54 (2009), 130–138; and other errata are also corrected. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

Some properties of edge intersection graphs of single-bend paths on a grid

In this paper we consider graphs G whose vertices can be represented as single-bend paths (i.e., paths with at most one turn) on a rectangular grid, such that two vertices are adjacent in G if and only if the corresponding paths share at least one edge of the grid. These graphs, called B 1 -EPG graphs, were first introduced in Golumbic et al. (2009) [13]. Here we show that the neighborhood of every vertex in a B 1 -EPG graph induces a weakly chordal graph. From this we conclude that the family F of B 1 -EPG graphs satisfies the Erdős–Hajnal property with ϵ ( F ) = 1 3 , i.e., that every B 1 -EPG graph on n vertices contains either a clique or a stable set of size at least n 1 3 . Finally we give a characterization of B 1 -EPG graphs among some subclasses of chordal graphs, namely chordal bull-free graphs, chordal claw-free graphs, chordal diamond-free graphs, and special cases of split graphs.

Recognizing Helly Edge-Path-Tree graphs and their clique graphs

We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs. The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices. Our result heavily relies on the notion of pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8–22] and on the observation that Helly Edge-Path-Tree matrices form a self-dual class of Helly matrices. Coupled to the notion of reduction presented in the paper, these facts are also exploited to reprove and slightly refine some known results for Edge-Path-Tree graphs.

On edge-intersection graphs of k-bend paths in grids

Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.

Some properties of edge intersection graphs of single bend paths on a grid

Abstract We consider the family of intersection graphs G of paths on a grid, where every vertex v in G corresponds to a single bend path Pv on a grid, and two vertices are adjacent in G if and only if the corresponding paths share an edge on the grid. We first show that these graphs have the Erdos-Hajnal property. Then we present some properties concerning the neighborhood of a vertex in these graphs, and finally we consider some subclasses of chordal graphs for which we give necessary and sufficient conditions to be edge intersection graphs of single bend paths in a grid.

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d ≥ 4 , in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ ( n 2 ) is needed. We also show several results related to the classes B k -EPG, where B k -EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n , the complete bipartite graph K m , n does not belong to B 2 m − 3 -EPG (it is known that this graph belongs to B 2 m − 2 -EPG); for any odd integer k we have B k -EPG ⫋ B k + 1 -EPG; there is no number k such that all graphs belong to B k -EPG; only 2 O ( k n log ( k n ) ) out of all the 2 n 2 labeled graphs with n vertices are in B k -EPG.

A superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree (EPT) graphs are intersection graphs of EPT matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. EPT graphs have polynomially many cliques [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinational Theory Series B 38 (1985) 8–22; C.L. Monma, V.K. Wey, Intersection graphs of paths in a tree, Journal of Combinational Theory Series B 41 (1986) 141–181]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. We extend this result to a proper superclass of EPT graphs.

Edge intersection graphs of single bend paths on a grid

AbstractWe combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let 𝒫 be a collection of nontrivial simple paths on a grid 𝒢. We define the edge intersection graph EPG(𝒫) of 𝒫 to have vertices which correspond to the members of 𝒫, such that two vertices are adjacent in EPG(𝒫) if the corresponding paths in 𝒫 share an edge in 𝒢. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EPG(𝒫) for some 𝒫 and 𝒢, and 〈𝒫,𝒢〉 is an EPG representation of G. We prove that every graph is an EPG graph. A turn of a path at a grid point is called a bend. We consider here EPG representations in which every path has at most a single bend, called B1‐EPG representations and the corresponding graphs are called B1‐EPG graphs. We prove that any tree is a B1‐EPG graph. Moreover, we give a structural property that enables one to generate non B1‐EPG graphs. Furthermore, we characterize the representation of cliques and chordless 4‐cycles in B1‐EPG graphs. We also prove that single bend paths on a grid have Strong Helly number 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

Intersection models of weakly chordal graphs

We first present new structural properties of a two-pair in various graphs. A two-pair is used in a well-known characterization of weakly chordal graphs. Based on these properties, we prove the main theorem: a graph G is a weakly chordal ( K 2 , 3 , 4 P 2 ¯ , P 2 ∪ P 4 ¯ , P 6 ¯ , H 1 , H 2 , H 3 )-free graph if and only if G is an edge intersection graph of subtrees on a tree with maximum degree 4. This characterizes the so called [4, 4, 2] graphs. The proof of the theorem constructively finds the representation. Thus, we obtain an algorithm to construct an edge intersection model of subtrees on a tree with maximum degree 4 for such a given graph. This is a recognition algorithm for [4, 4, 2] graphs.

Integrality properties of edge path tree families

An Edge Path Tree (EPT) family is a family whose members are edge sets of paths in a tree. Relying on the notion of Pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8–22], we characterize Ideal and Mengerian EPT families. In particular, we show that an EPT family is Ideal if and only if it is Mengerian. If, in addition, the EPT family is uniform, then it is Ideal if and only if it is Unimodular. The latter equivalence generalizes the well-known fact that the edge set of a graph is an Ideal clutter if and only if the graph is bipartite.

Edge intersection graphs of linear 3-uniform hypergraphs

Let L 3 l be the class of edge intersection graphs of linear 3-uniform hypergraphs. It is known that the problem of recognition of the class L 3 l is NP-complete. We prove that this problem is polynomially solvable in the class of graphs with minimum vertex degree ≥ 10 . It is also proved that the class L 3 l is characterized by a finite list of forbidden induced subgraphs in the class of graphs with minimum vertex degree ≥ 16 .

Clique-coloring UE and UEH graphs

Abstract We consider the clique-coloring problem, that is, coloring the vertices of a given graph such that no maximal clique of size at least two is monocolored. More specifically, we investigate the problem of giving a class of graphs, to determine if there exists a constant C such that every graph in this class is C-clique-colorable. We consider the classes of UE and UEH graphs. A graph G is called an UE graph if it is the edge intersection graph of a family of paths in a tree. If this family satisfies the Helly Property we say that G is an UEH graph. We show that every UEH graph is 3-clique-colorable. Moreover our proof is constructive and provides a polynomial-time algorithm. We also describe, for each k ≥ 2 , an UE graph that is not k-clique-colorable. The UE graphs form one of the few known classes for which the clique-chromatic number is unbounded.

The [formula omitted]-edge intersection graphs of paths in a tree

We consider a generalization of edge intersection graphs of paths in a tree. Let P be a collection of nontrivial simple paths in a tree T . We define the k -edge ( k ⩾ 1 ) intersection graph Γ k ( P ) , whose vertices correspond to the members of P , and two vertices are joined by an edge if the corresponding members of P share k edges in T . An undirected graph G is called a k -edge intersection graph of paths in a tree, and denoted by k -EPT, if G = Γ k ( P ) for some P and T . It is known that the recognition and the coloring of the 1-EPT graphs are NP-complete. We extend this result and prove that the recognition and the coloring of the k -EPT graphs are NP-complete for any fixed k ⩾ 1 . We show that the problem of finding the largest clique on k -EPT graphs is polynomial, as was the case for 1-EPT graphs, and determine that there are at most O ( n 3 ) maximal cliques in a k -EPT graph on n vertices. We prove that the family of 1-EPT graphs is contained in, but is not equal to, the family of k -EPT graphs for any fixed k ⩾ 2 . We also investigate the hierarchical relationships between related classes of graphs, and present an infinite family of graphs that are not k -EPT graphs for every k ⩾ 2 . The edge intersection graphs are used in network applications. Scheduling undirected calls in a tree is equivalent to coloring an edge intersection graph of paths in a tree. Also assigning wavelengths to virtual connections in an optical network is equivalent to coloring an edge intersection graph of paths in a tree.

Representing edge intersection graphs of paths on degree 4 trees

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P , denoted by EPT ( P ) , has vertex set that corresponds to the members of P , and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G = EPT ( P ) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement G ¯ possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.

On the strong p-Helly property

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8–22]. Independently, other authors [A. Bretto, S. Ubéda, J. Žerovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55–57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216–220, W.D. Wallis, Guo-Hui Zhang, On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187–193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p = 2 , this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216–220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.

Edge intersection graphs of linear 3-uniform hypergraphs

Abstract Let L 3 l be the class of edge intersection graphs of linear 3-uniform hypergraphs. The problem of recognizing G ∈ L 3 l is NP-complete. Denote by δ ALG the minimal integer such that the problem G ∈ L 3 l is polynomially solvable in the class of graphs G with the minimal vertex degree δ ( G ) ≥ δ ALG and by δ FIS the minimal integer such that L 3 l can be characterized by a finite list of forbidden induced subgraphs in the class of graphs G with δ ( G ) ≥ δ FIS . It is proved that δ ALG ≤ 10 and δ FIS ≤ 16 .

Representations of Edge Intersection Graphs of Paths in a Tree

Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.