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.     

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.     

Packing paths in planar graphs

A generalization of P. Seymour's theorem on planar integral 2-commodity flows is given when the underlying graphG together with the demand graphH (a graph having edges that connect the corresponding terminal pairs) form a planar graph and the demand edges are on two faces ofG.     

On paths in planar graphs

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y and containing any specified edge other than xy. © 1997 John Wiley & Sons, Inc.     

Monotone paths in ordered graphs

LetV _fin andE _fin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesV _inf andE _inf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveE _inf ⫋V _inf ⫋V _fin ⫋E _fin. Finally we consider labellings by positive integers and characterize the class corresponding toV _inf.     

Disjoint paths in sparse graphs

We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg, Vazirani and Yannakakis [N. Garg, V.V. Vazirani, M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees, Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of trees. We also introduce thek-edge-outerplanar graphs, a class of planar graphs with arbitrary (but bounded) tree-width that generalizes the cacti, and show that the integrality gap of the maximum edge-disjoint paths problem is bounded in these graphs.     

Disjoint shortest paths in graphs

It is an interesting problem that how much connectivity ensures the existence ofn disjoint paths joining givenn pairs of vertices, but to get a sharp bound seems to be very difficult. In this paper, we study how muchgeodetic connectivity ensures the existence ofn disjointgeodesics joining givenn pairs of vertices, where a graph is calledk-geodetically connected if the removal of anyk−1 vertices does not change the distance between any remaining vertices.     

Intersection theory for graphs

An intersection theory developed by the author for matroids embedded in uniform geometries is applied to the case when the ambient geometry is the lattice of partitions of a finite set so that the matroid is a graph. General embedding theorems when applied to graphs give new interpretations to such invariants as the dichromate of Tutte. A polynomial in n + 1 variables, the polychromate, is defined for graphs with n vertices. This invariant is shown to be strictly stronger than the dichromate, it is edge-reconstructible and can be calculated for proper graphs from the polychromate of the complementary graph. By using Tutte's construction for codichromatic graphs (J. Combinatorial Theory 16 (1974), 168–174), copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper.     

Intersection graphs of proper subtrees of unicyclic graphs

A graph is fraternally oriented iff for every three vertices u, ν, w the existence of the edges u → w and ν → w implies that u and ν are adjacent. A directed unicyclic graph is obtained from a unicyclic graph by orienting the unique cycle clockwise and by orienting the appended subtrees from the cycle outwardly. Two directed subtrees s, t of a directed unicyclic graph are proper if their union contains no (directed or undirected) cycle and either they are disjoint or one of them s has its root r(s) in t and contains all the successors of r(s) in t. In the present paper we prove that G is an intersection graph of a family of proper directed subtrees of a directed unicyclic graph iff it has a fraternal orientation such that for every vertex ν, G(Γⁱⁿν) is acyclic and G(Γ^outν) is the transitive closure of a tree. We describe efficient algorithms for recognizing when such graphs are perfect and for testing isomorphism of proper circular-arc graphs.     

Hamiltonian paths in infinite graphs

A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ ₁ ¹ -complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case. Parts of this research were carried out during a visit to IBM T.J. Watson Research Center, Hawthorne, NY, in the Summer of 1990. The author holds the William Sussman Professorial Chair in Mathematics.     

Hamiltonian paths in oriented graphs

A short proof is given of Meyniel's theorem on Hamiltonian cycles in oriented graphs. Analogous conditions are obtained for a graph to be Hamiltonianconnected.     

Edge-disjoint paths in planar graphs

Given a planar graph G = (V, E), find k edge-disjoint paths in G connecting k pairs of terminals specified on the outer face of G. Generalizing earlier results of Okamura and Seymour (J. Combin. Theory Ser. B31 (1981), 75–81) and of the author (Combinatorica2, No. 4 (1982), 361–371), we solve this problem when each node of G not on the outer face has even degree. The solution involves a good characterization for the solvability and the proof gives rise to an algorithm of complexity O(|V|³log|V|). In particular, the integral multicommodity flow problem is proved to belong to the problem class P when the underlying graph is outerplanar.     

Hamiltonian paths in Cayley graphs

The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log₂|G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3-regular Hamiltonian expanders.     

Intersection diagrams of distance-biregular graphs

We shall investigate distance-biregular graphs by means of intersection diagrams. First we give an alternate proof of a theorem which was obtained by Mohar and Shawe-Taylor in (J. Combin. Theory Ser. B 37 (1984), 90–100). Next we give some results on distance-biregular graphs of girth g ≡ 0 (mod 4).     

Line graphs of bipartite graphs with Hamiltonian paths

It is shown that, if t is an integer ≥3 and not equal to 7 or 8, then there is a unique maximal graph having the path P_t as a star complement for the eigenvalue ?2. The maximal graph is the line graph of K_m,m if t = 2m?1, and of K_m,m+1 if t = 2m. This result yields a characterization of L(G ) when G is a (t + 1)‐vertex bipartite graph with a Hamiltonian path. The graphs with star complement P_r ∪ P_s or P_r ∪ C_s for ?2 are also determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 137–149, 2003