2-disjoint Path Research Articles

Paired 2-disjoint path covers of k-ary n-cubes under the partitioned edge fault model

The k-ary n-cube Qnk serves as an indispensable interconnection network in the design of data center networks, network-on-chips, and parallel computing systems since it possesses numerous attractive properties. In these parallel architectures, the paired (or unpaired) many-to-many m-disjoint path cover (m-DPC) plays a significant role in message transmission. Nevertheless, the construction of m-DPC is severely obstructed by large-scale edge faults due to the rapid growth of the system scale. In this paper, we investigate the existence of paired 2-DPC in Qnk under the partitioned edge fault (PEF) model, which is a novel fault model for enhancing the networks' fault-tolerance related to path embedding problem. We exploit this model to evaluate the edge fault-tolerance of Qnk when a paired 2-DPC is embedded into Qnk. Compared to the other known works, our results can help Qnk to achieve large-scale edge fault-tolerance.

Long directed detours: Reduction to 2-Disjoint Paths

In the Longest(s,t)-Detour problem, we look for an (s,t)-path that is at least k vertices longer than a shortest one. We study the parameterized complexity of Longest(s,t)-Detour when parameterized by k: this falls into the research paradigm of ‘parameterization above guarantee’. Whereas the problem is known to be fixed-parameter tractable (FPT) on undirected graphs, the status of Longest(s,t)-Detour on directed graphs remains highly unclear: it is not even known to be solvable in polynomial time for k=1. Recently, Fomin et al. made progress in this direction by showing that the problem is FPT on every class of directed graphs where the 3-Disjoint Paths problem is solvable in polynomial time. We improve upon their result by weakening this assumption: we show that only a polynomial-time algorithm for 2-Disjoint Paths is required.

Paired many-to-many 2-disjoint path covers in meshes

Paired many-to-many 2-disjoint path covers in meshes

Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph BPn is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let {u,v} and {x,y} be any two pairs of distinct vertices of BPn for n≥4. We show that there are u−v and x−y paths whose vertices partition the vertex set of BPn even if BPn has up to n−4 faulty elements. On the other hand, for every n≥3 there is a set of n−2 faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.

The disjoint path cover in the data center network HSDC with prescribed vertices in each path

The n-dimensional HSDC with the logic graph Hn is one of the most attractive server-centric data center networks for high incremental scalability. The routing design in network topology is very important for information transmission. In particular, the application of disjoint path covers can solve various problems such as program code optimization and mapping parallel programs to parallel structures. Hn consists of 2n cliques and each two vertices in a clique are adjacent. For any vertex sets {x,u} and {y,v} in W and B, respectively, and any two disjoint vertex sets A1,A2 in some clique, where (W,B) is a bipartition of Hn, in this paper, we prove that, when |Ai|≠1 and Ai∩{x,y,u,v}=∅ for i=1,2, if |A1∪A2|≤n−3, then there exists an (x,y)-path P1 and a (u,v)-path P2 in Hn passing through A1 and A2, respectively, such that P1 and P2 have no common vertices and P1∪P2 contains all vertices of Hn. Then P1∪P2 is called the 2-disjoint path cover in Hn and we propose the efficient algorithms for the path design. Furthermore, we prove that our main result is optimal.

Characterization of interval graphs that are paired 2-disjoint path coverable

Characterization of interval graphs that are paired 2-disjoint path coverable

Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubest

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

Characterization of interval graphs that are unpaired 2-disjoint path coverable

Given disjoint source and sink sets, S={s1,…,sk} and T={t1,…,tk}, in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which Pi is a path from source si to some sink tj. In terms of a generalized scattering number, named an r-scattering number, we characterize interval graphs that have an unpaired 2-disjoint path cover joining S and T for any possible configurations of source and sink sets S and T of size 2 each. Also, it is shown that the r-scattering number of an interval graph can be computed in polynomial time.

Finding a path with two labels forbidden in group-labeled graphs

The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s – t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s – t path in a Z 3 -labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s – t paths in a group-labeled graph or not, and finding s – t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s – t paths, which is the main technical contribution of this paper.

Paired 2-disjoint path covers of multi-dimensional torus networks with 2n − 3 faulty edges

The n-dimensional torus T(k1,k2,…,kn) (including the k-ary n-cube Qnk) is one of the most popular interconnection networks. A paired k-disjoint path cover (paired k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all vertices of the graph. In this paper, we consider the paired 2-DPC problem of n-dimensional torus. Assuming ki≥3 for i=1,2,…,n, with at most one ki being even, then T(k1,k2,…,kn) with at most 2n−3 faulty edges always has a paired 2-DPC. And the upper bound 2n−3 of edge faults tolerated is optimal. The result is a supplement of the results of Chen [3] and [4].

A linear-time algorithm for finding a paired 2-disjoint path cover in the cube of a connected graph

For a connected graph G=(V(G),E(G)) and two disjoint subsets of V(G) A={α1,…,αk} and B={β1,…,βk}, a paired (many-to-many) k-disjoint path cover of G joining A and B is a vertex-disjoint path cover {P1,…,Pk} such that Pi is a path from αi to βi for 1≤i≤k. In the recent paper, Park and Ihm (2014) presented a necessary and sufficient condition for a paired 2-disjoint path cover joining two vertex sets to exist in the cube of a connected graph. In this paper, we propose an O(|V(G)|+|E(G)|)-time algorithm that actually finds such a paired 2-disjoint path cover. In particular, we show that, in order to build a desired disjoint path cover, it is sufficient to consider only the edges of a carefully selected spanning tree of the graph and at most one additional edge not in the tree, which allows an efficient linear-time algorithm.

Paired 2-disjoint path covers of faulty k-ary n-cubes

A paired k-disjoint path cover of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all vertices of the graph. The k-ary n-cube Qnk is one of the most popular interconnection networks. In this paper, we consider the problem of paired 2-disjoint path covers of the k-ary n-cube Qnk (odd k≥5) with faulty elements (vertices and/or edges) and obtain the following result. Let F be a set of faulty elements in Qnk (odd k≥5) with |F|≤2n−4, and {a,b} and {c,d} be its any two pairs of non-faulty vertices. Then the graph Qnk−F contains vertex-disjoint a−b path and c−d path that cover its all non-faulty vertices, and the upper bound 2n−4 of faults tolerated is nearly optimal. We also show that Cartesian product Qnk×P with at most 2n−2 faulty elements is Hamiltonian connected, where P is a path, n≥2 and odd k≥5.

Paired 2-disjoint path covers of multidimensional torus networks with faulty edges

A paired k-disjoint path cover (paired k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all vertices of the graph. Clearly, the paired k-DPC is stronger than Hamiltonian-connectivity. The n-dimensional torus T(k1,k2,…,kn) (including the k-ary n-cube Qnk) is one of the most popular interconnection networks. In this paper, we obtain the following results. (1) Assume even ki≥4 for i=1,2,…,n. Let T=T(k1,k2,…,kn) be a bipartite torus and F be a set of faulty edges with |F|≤2n−3. Given any four vertices s1,t1,s2 and t2, such that each partite set contains two vertices. Then the graph T−F has a paired 2-DPC consisting of s1−t1 path and s2−t2 path. And the upper bound 2n−3 of edge faults tolerated is optimal. The result is a generalization of the result of Park et al. concerning the case of n=2[17]. (2) Assume ki≥3 for i=1,2,…,n, with at most one ki being even. Let T=T(k1,k2,…,kn) be a torus and F be a set of faulty edges with |F|≤2n−4. Then the graph T−F has a paired 2-DPC. And the upper bound 2n−4 of edge faults tolerated is nearly optimal. The result is a generalization of the result of Park concerning the case of n=2[16]. Our brief proofs are based on a technique that is of interest and may find some applications.

Many-to-many two-disjoint path covers in cylindrical and toroidal grids

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex sets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. The many-to-many k-disjoint path cover is classified as paired if each source in S is further required to be paired with a specific sink in T, or unpaired otherwise. In this paper, we first establish a necessary and sufficient condition for a bipartite cylindrical grid to have a paired many-to-many 2-disjoint path cover joining S and T. Based on this characterization, we then prove that, provided the set S∪T contains the equal numbers of vertices from different parts of the bipartition, the bipartite cylindrical grid always has an unpaired many-to-many 2-disjoint path cover. Additionally, we show that such balanced vertex sets also guarantee the existence of a paired many-to-many 2-disjoint path cover for any bipartite toroidal grid even if an arbitrary edge is removed.

Disjoint path covers in cubes of connected graphs

Given a graph G, and two vertex sets S and T of size k each, a many-to-many k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T, or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient condition for the cube of a connected graph to have a paired 2-disjoint path cover. Then, a corresponding condition for the unpaired type of 2-disjoint path cover problem is immediately derived. It is also shown that these results can easily be extended to determine if the cube of a connected graph has a hamiltonian path from a given vertex to another vertex that passes through a prescribed edge.

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

A paired many-to-many k-disjoint path cover (paired k-DPC for short) of a graph is a set of k vertex-disjoint paths joining k distinct source-sink pairs that altogether cover every vertex of the graph. We consider the problem of constructing paired 2-DPC’s in an m-dimensional bipartite HL-graph, Xm, and its application in finding the longest possible paths. It is proved that every Xm, m⩾4, has a fault-free paired 2-DPC if there are at most m−3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition. Furthermore, every Xm, m⩾4, has a paired 2-DPC in which the two paths have the same length if each source-sink pair is balanced. Using 2-DPC properties, we show that every Xm, m⩾3, with either at most m−2 faulty edges or one faulty vertex and at most m−3 faulty edges is strongly Hamiltonian-laceable.

The 2-path-bipanconnectivity of hypercubes

In this paper, we introduce the concept of the k-path-(bi)panconnectivity of (bipartite) graphs. It is a generalization of the (bi)panconnectivity and of the paired many-to-many k-disjoint path cover. The 2-path-bipanconnectivity with only one exception of the n-cube Qn (n⩾4) is proved. Precisely, the following result is obtained: In an n-cube with n⩾4 given any four vertices u1, v1, u2, v2 such that two of them are in one partite set and the another two are in the another partite set. Let s=t=5 if C=u1u2v1v2 is a cycle of length 4, and s=d(u1,v1)+1 and t=d(u2,v2)+1 otherwise, where d(u,v) denotes the distance between two vertices u and v. And let i and j be any two integers such that both i−s⩾0 and j−t⩾0 are even with i+j⩽2n. Then there exist two vertex-disjoint (u1,v1)-path P and (u2,v2)-path R with ∣V(P)∣=i and ∣V(R)∣=j. As consequences, many properties of hypercubes, such as bipanconnectivity, bipanpositionable bipanconnectivity [18], bipancycle-connectivity [12], two internally disjoint paths with two given lengths, and the 2-disjoint path cover with a path of a given length [21], follow from our result.

Linear time algorithms for two disjoint paths problems on directed acyclic graphs

We present an algorithm that, given two vertices s1 and s2 of a directed acyclic graph, constructs in linear time a data structure using linear space that, for each pair (u,v) of two vertices u and v, in constant time can output a tuple (s1,t1,s2,t2) with {t1,t2}={u,v} such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. The data structure is simpler than such a data structure for general directed graphs that can be obtained from results of Georgiadis and Tarjan. Disjoint can mean vertex- as well as edge-disjoint. As an application and main result we show that the data structure can be used to solve the 2-disjoint paths problem on directed acyclic graphs optimally, i.e., in linear time.

Solving the 2-Disjoint Paths Problem in Nearly Linear Time

Given four distinct vertices s1,s2,t1, and t2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p1 from s1 to t1 and p2 from s2 to t2, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the 2-vertex- and the 2-edge-disjoint paths problem on dags.