Edge-disjoint Paths Of Length Research Articles

Number of vertices of degree three in spanning 3-trees in square graphs

In this paper, we show that the square graph of a tree T has a spanning tree of maximum degree at most three and with at most max{0,∑x∈W≥3(T)(tT(x)−2)−2} vertices of degree three, where W≥3(T)={x∈V(T): there are at least three edge-disjoint paths of length at least two that start x} and tT(x) is the number of edge-disjoint paths with length at least two that start at a vertex x.

Integer programming formulations for thek-edge-connected 3-hop-constrained network design problem

In this article, we study the k-edge-connected L-hop-constrained network design problem. Given a weighted graph G = (V,E), a set D of pairs of nodes, two integers L ≥ 2 and k ≥ 2, the problem consists in finding a minimum weight subgraph of G containing at least k edge-disjoint paths of length at most L between every pair {s, t } ∈ D. We consider the problem in the case where L = 2, 3 and |D| ≥ 2. We first discuss integer programming formulations introduced in the literature. Then, we introduce new integer programming formulations for the problem that are based on the transformation of the initial undirected graph into directed layered graphs. We present a theoretical comparison of these formulations in terms of LP-bound. Finally, these formulations are tested using CPLEX and compared in a computational study for k = 3, 4, 5.

Length 3 Edge-Disjoint Paths Is NP-Hard

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and in this note we show that this problem is NP-hard. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard.

The [formula omitted] edge-disjoint 3-hop-constrained paths polytope

Given a graph G with two distinguished nodes s and t , a cost on each edge of G and two fixed integers k ≥ 2 , L ≥ 2 , the k edge-disjoint L -hop-constrained paths problem is to find a minimum cost subgraph of G such that between s and t there are at least k edge-disjoint paths of length at most L . In this paper we consider this problem from a polyhedral point of view. We give an integer programming formulation for the problem and discuss the associated polytope. In particular, we show that when L = 3 and k ≥ 2 , the linear relaxation of the associated polytope, given by the trivial, the s t -cut and the so-called L -path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2 , 3 and k ≥ 1 . This generalizes the results of Huygens et al. (2004) [1] for k = 2 and L = 2 , 3 and those of Dahl et al. (2006) [2] for L = 2 and k ≥ 2 . This also proves a conjecture in [1].

Euler Tours of Maximum Girth in K2n+1 and K2n,2n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g E (G)= maxg(T) where the maximum is taken over all Euler tours of G. We prove that g E (K 2 n ,2 n )=4n?4 and 2n?3≤g E (K 2 n +1)≤2n?1 for any n?2. We also show that g E (K 7)=4. We use these results to prove the following: 1)The graph K 2 n ,2 n can be decomposed into edge disjoint paths of length k if and only if k≤4n?1 and the number of edges in K 2 n ,2 n is divisible by k. 2)The graph K 2 n +1 can be decomposed into edge disjoint paths of length k if and only if k≤2n and the number edges in K 2 n +1 is divisible by k.

Two Edge-Disjoint Hop-Constrained Paths and Polyhedra

Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a fixed integer L \geq 2, the two edge-disjoint hop-constrained paths problem is to find a minimum cost subgraph such that between s and t there exist at least two edge-disjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem when L = 2,3. An extension of this result to the more general case where the number of required paths is arbitrary and L = 2,3 is also given. We discuss the associated polytope, P(G,L), for L = 2,3. In particular, we show in this case that the linear relaxation of P(G,L), Q(G,L), given by the trivial, the st-cut, and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2,3. We also give necessary and sufficient conditions for these inequalities to define facets of P(G,L) for L \geq 2 when G is complete. We finally investigate the dominant of P(G,L) and give a complete description of this polyhedron for L \geq 2 when P(G,L) = Q(G,L).

Pk+1–Decompositions of Eulerian Graphs: Complexity and Some Solvable Cases

We consider the problem of PMk+1-decomposition of a simple eulerian graph G, that is, decomposition of G into edge disjoint paths of length k. We show that the problem of deciding whether there exists a Pk+1 - decomposition of an eulerian simple graph is NP-complete, for every k = 3. However we find some new classes of graphs where the problem of P4-decomposition can be solved polynomially. We show that an eulerian simple graph G on 3m = 6 edges admits a P4-decomposition if G has no cut vertex v such that exactly one of the components in the graph G - ? has two vertices. In particular, this implies that a 2-connected eulerian simple graph G on 3m = 6 edges admits a P4 -decomposition. © 2003.

On the robustness of interconnections in random graphs: a symbolic approach

Graphs are models of communication networks. This paper applies symbolic combinatorial techniques in order to characterize the interplay between two parameters of a random graph, namely its density (the number of edges in the graph) and its robustness to link failures. Here, robustness means multiple connectivity by short disjoint paths: a triple (G,s,t), where G is a graph and s,t are designated vertices, is called ℓ-robust if s and t are connected via at least two edge-disjoint paths of length at most ℓ. We determine the expected number of ways to get from s to t via two edge-disjoint paths of length ℓ in the classical random graph model G n,p by means of “symbolic” combinatorial methods. We then derive bounds on related threshold probabilities as functions of ℓ and n.