EPT Graphs Research Articles

Clique coloring EPT graphs on bounded degree trees

Clique coloring EPT graphs on bounded degree trees

Minimum Number of Bends of Paths of Trees in a Grid Embedding

We are interested in embedding trees T with ∆(T) ≤ 4 in a rectangular grid, such that the vertices of T correspond to grid points, while edges of T correspond to non-intersecting straight segments of the grid lines. Such embeddings are called straight models. While each edge is represented by a straight segment, a path of T is represented in the model by the union of the segments corresponding to its edges, which may consist of a path in the model having several bends. The aim is to determine a straight model of a given tree T minimizing the maximum number of bends over all paths of T. We provide a quadratic-time algorithm for this problem. We also show how to construct straight models that have k as its minimum number of bends and with the least number of vertices possible. As an application of our algorithm, we provide an upper bound on the number of bends of EPG models of VPT ∩ EPT graphs.

Relationship among B1-EPG, VPT and EPT graphs classes

Relationship among <i>B₁</i>-EPG, VPT and EPT graphs classes

EPT graphs on bounded degree trees

EPT graphs on bounded degree trees

Constant threshold intersection graphs of orthodox paths in trees

A graph G belongs to the class ORTH[h,s,t] for integers h, s, and t if there is a pair (T,S), where T is a tree of maximum degree at most h, and S is a collection (Su)u∈V(G) of subtrees Su of maximum degree at most s of T, one for each vertex u of G, such that, for every vertex u of G, all leaves of Su are also leaves of T, and, for every two distinct vertices u and v of G, the following three properties are equivalent:(i) u and v are adjacent.(ii) Su and Sv have at least t nodes in common.(iii) Su and Sv share a leaf of T.The class ORTH[h,s,t] was introduced by Jamison and Mulder.Here we focus on the case s=2, which is closely related to the well-known VPT and EPT graphs. We collect general properties of the graphs in ORTH[h,2,t], and provide a characterization in terms of tree layouts. Answering a question posed by Golumbic, Lipshteyn, and Stern, we show that ORTH[h+1,2,t]∖ORTH[h,2,t] is non-empty for every h≥3 and t≥3. We derive decomposition properties, which lead to efficient recognition algorithms for the graphs in ORTH[h,2,2] for every h≥3. Finally, we give a complete description of the graphs in ORTH[h,2,2], and show that the graphs in ORTH[3,2,3] are line graphs of planar graphs.

Intersection Graphs of Orthodox Paths in Trees

We study the graph classes ORTH[h,s,t] introduced by Jamison and Mulder, and focus on the case s=2, which is closely related to the well-known VPT and EPT graphs. We collect general properties of the graphs in ORTH[h,2,t], and provide a characterization in terms of tree layouts. Answering a question posed by Golumbic, Lipshteyn, and Stern, we show that ORTH[h+1,2,t] \\ ORTH[h,2,t] is non-empty for every h≥3 and t≥3. We derive decomposition properties, which lead to efficient recognition algorithms for the graphs in ORTH[h,2,2] for every h≥3. Finally, we show that the graphs in ORTH[3,2,3] are line graphs of planar graphs.

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.

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.

A necessary condition for EPT graphs and a new family of minimal forbidden subgraphs

A necessary condition for EPT graphs and a new family of minimal forbidden subgraphs

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An ( h , s , t ) -representation of a graph G consists of a collection of subtrees of a tree T , where each subtree corresponds to a vertex in G , such that (i) the maximum degree of T is at most h , (ii) every subtree has maximum degree at most s , (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T . The class of graphs that have an ( h , s , t ) -representation is denoted by [ h , s , t ] . It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below. In this paper, we investigate the class of [ h , 2 , t ] graphs, i.e., the intersection graphs of paths in a tree. The [ h , 2 , 1 ] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211–227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151–159], and [ h , 2 , 2 ] graphs are known as the EPT graphs. We consider variations of [ h , 2 , t ] by three main parameters: h , t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [ h , 2 , t ] and orthodox- [ h , 2 , t ] graphs for varied values of h and t .

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.

Twenty Years of EPT Graphs: From Haifa to Rostock

Twenty Years of EPT Graphs: From Haifa to Rostock

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.

Edge and vertex intersection of paths in a tree

In this paper we continue the investigation of the class of edge intersection graphs of a collection of paths in a tree (EPT graphs) where two paths edge intersect if they share an edge. The class of EPT graphs differs from the class known as path graphs, the latter being the class of vertex intersection graphs of paths in a tree. A characterization is presented here showing when a path graph is an EPT graph. In particular, the classes of path graphs and EPT graphs coincide on trees all of whose vertices have degree at most 3. We then prove that it is an NP-complete problem to recognize whether a graph is an EPT graph.

The edge intersection graphs of paths in a tree

The class of edge intersection graphs of a collection of paths in a tree (EPT graphs) is investigated, where two paths edge intersect if they share an edge. The cliques of an EPT graph are characterized and shown to have strong Helly number 4. From this it is demonstrated that the problem of finding a maximum clique of an EPT graph can be solved in polynomial time. It is shown that the strong perfect graph conjecture holds for EPT graphs. Further complexity results follow from the observation that every line graph is an EPT graph. The class of EPT graphs is equivalent to the class of fundamental cycle graphs.