Vertex-disjoint Paths Research Articles

Fault-Tolerant Maximal Local-Connectivity on Cayley Graphs Generated by Transpositions

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.

Nontrivial path covers of graphs: existence, minimization and maximization

Let G be a graph and $${{{\mathcal {P}}}}$$ be a set of pairwise vertex-disjoint paths in G. We say that $${{\mathcal {P}}}$$ is a path cover if every vertex of G belongs to a path in $${{\mathcal {P}}}$$. In the minimum path cover problem, one wishes to find a path cover of minimum cardinality. In this problem, known to be $${\textsc {NP}}$$-hard, the set $${{\mathcal {P}}}$$ may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we show that the corresponding existence problem can be reduced to a matching problem. This reduction gives, in polynomial time, a certificate for both the yes-answer and the no-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the cover. We show that, the minimization problem on feasible instances is computationally equivalent to the minimum path cover problem: their optimum values coincide and they have the same approximation threshold. We show that the maximization problem can be solved in polynomial time. We also consider a weighted version of the path cover problem, in which we seek a path cover with minimum or maximum total weight (the number of paths do not matter), and we show that while the first is polynomial, the second is NP-hard, but admits a constant-factor approximation algorithm. We also describe a linear-time algorithm on (weighted) trees, and mention results for graphs with bounded treewidth.

The path cover problem: Formulation and a hybrid metaheuristic

This paper introduces the path cover problem (PCP) that is a variant of the capacitated vehicle routing problem (CVRP). In contrast to CVRP, a vehicle route in PCP involves a set of vertex-disjoint paths where a vehicle starts at a customer and finishes at another customer without traveling to the depot. PCP is defined to find a set of service paths to serve a set of customers that aiming to minimize the total traveling cost and hiring cost. This paper develops a hybrid approach that integrates variable neighborhood search (VNS) and tabu search (TS) to solve the problem. This algorithm presents an approach to explore the solution space by utilizing multi-set neighborhood and applying the systematic changing neighborhood of VNS. TS is incorporated into the algorithm to guide the search toward diverse regions. The computational results indicate that the proposed algorithm efficiently solves PCP.

Efficient algorithms for measuring the funnel-likeness of DAGs

Funnels are a new natural subclass of DAGs. Intuitively, a DAG is a funnel if every source-sink path can be uniquely identified by one of its arcs. Funnels are an analog to trees for directed graphs that is more restrictive than DAGs but more expressive than in-/out-trees. Computational problems such as finding vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs while on funnels they become solvable in polynomial time. Our main focus is the algorithmic complexity of finding out how funnel-like a given DAG is. To this end, we study the NP-hard problem of computing the arc-deletion distance to a funnel of a given DAG. We develop efficient exact and approximation algorithms for the problem and test them on synthetic random graphs and real-world graphs.

Minimum Path Cover in Quasi-claw-Free Graphs

A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., $$p(G)=\min \{|{\mathcal {P}}|:$$$${\mathcal {P}}$$ is a path cover of $$G\}.$$ A graph G is quasi-claw-free if for any two vertices x, y with $$d(x,y)=2$$ in G, there exists a vertex u with $$u\in N(x)\cap N(y)$$ and $$N(u)\subseteq N[x]\cup N[y].$$ In this paper, we prove that for any quasi-claw-free graph G of order n, if $$\sigma _{k+1}(G)\ge {n-k}$$ for a positive integer k, then $$p(G)\le k,$$ where $$\sigma _{k+1}(G)$$ is the minimum degree sum of an independent set with $$k+1$$ vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.

Characterizing the Difference Between Graph Classes Defined by Forbidden Pairs Including the Claw

For two graphs A and B, a graph G is called $$\{A,B\}$$-free if G contains neither A nor B as an induced subgraph. Let $$P_{n}$$ denote the path of order n. For nonnegative integers k, $$\ell $$ and m, let $$N_{k,\ell ,m}$$ be the graph obtained from $$K_{3}$$ and three vertex-disjoint paths $$P_{k+1}$$, $$P_{\ell +1}$$, $$P_{m+1}$$ by identifying each of the vertices of $$K_{3}$$ with one endvertex of one of the paths. Let $$Z_{k}=N_{k,0,0}$$ and $$B_{k,\ell }=N_{k,\ell ,0}$$. Bedrossian characterized all pairs $$\{A,B\}$$ of connected graphs such that every 2-connected $$\{A,B\}$$-free graph is Hamiltonian. All pairs appearing in the characterization involve the claw ($$K_{1,3}$$) and one of $$N_{1,1,1}$$, $$P_{6}$$ and $$B_{1,2}$$. In this paper, we characterize connected graphs that are (i) $$\{K_{1,3},Z_{2}\}$$-free but not $$B_{1,1}$$-free, (ii) $$\{K_{1,3},B_{1,1}\}$$-free but not $$P_{5}$$-free, or (iii) $$\{K_{1,3},B_{1,2}\}$$-free but not $$P_{6}$$-free. The third result is closely related to Bedrossian’s characterization. Furthermore, we apply our characterizations to some forbidden pair problems.

A polynomial-time algorithm of finding a minimum k-path vertex cover and a maximum k-path packing in some graphs

For a graph G and a positive integer k, a subset C of vertices of G is called a k-path vertex cover if C intersects all paths of k vertices in G. The cardinality of a minimum k-path vertex cover is denoted by $$\beta _{P_k}(G)$$ . For a graph G and a positive integer k, a subset M of pairwise vertex-disjoint paths of k vertices in G is called a k-path packing. The cardinality of a maximum k-path packing is denoted by $$\mu _{P_k}(G)$$ . In this paper, we describe some graphs, having equal values of $$\beta _{P_k}$$ and $$\mu _{P_k}$$ , for $$k \ge 5$$ , and present polynomial-time algorithms of finding a minimum k-path vertex cover and a maximum k-path packing in such graphs.

Reliability Analysis for Disjoint Paths

Our contemporary society survives on the services provided by several network infrastructures, such as communication, power, and transportation, so their reliability should be accurately evaluated from various aspects. While past studies defined network reliability in terms of connectivity, some of the network connections that have been established may suffer from insufficient resources, i.e., vertices may be connected on a network but flows might fail due to resource contention. As a first step to address resource protection, this paper introduces path disjointness to the field of network reliability analysis, i.e., we evaluate the probability that given terminals are connected via edge- or vertex-disjoint paths. In addition, we also deal with identifying critical links under path disjointness. Since network reliability analysis is a computationally tough problem, we propose efficient algorithms utilizing the data structure of binary decision diagrams. Numerical experiments show that our method scales up to a network with 189 links. We show that network reliability and the criticality of links are greatly dependent on path disjointness; this validates the importance of the proposed method.

Computing Vertex-Disjoint Paths in Large Graphs Using MAOs

We consider the problem of computing $$k \in {\mathbb {N}}$$ internally vertex-disjoint paths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since 42 years, $$O(\min \{k,\sqrt{n}\}m)$$ for each pair by using traditional flow-based methods. The restriction of our vertex pairs comes from the machinery of maximal adjacency orderings (MAOs). Henzinger showed for every MAO and every $$1 \le k \le \delta $$ (where $$\delta $$ is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last $$\delta -k+2$$ vertices of this MAO. Later, Nagamochi generalized this result by using the machinery of mixed connectivity. Both results are however inherently non-constructive. We present the first algorithm that computes these k internally vertex-disjoint paths in linear time $$O(n+m)$$, which improves the previously best time $$O(\min \{k,\sqrt{n}\}m)$$. Due to the linear running time, this algorithm is suitable for large graphs. The algorithm is simple, works directly on the MAO structure, and completes a long history of purely existential proofs with a constructive method. We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms.

The Local Structure of Claw-Free Graphs Without Induced Generalized Bulls

In this paper, we show the following: Let G be a connected claw-free graph such that G has a connected induced subgraph H that has a pair of vertices $$\{v_{1}, v_{2}\}$$ of degree one in H whose distance is $$d + 2$$ in H. Then H has an induced subgraph F, which is isomorphic to $$B_{i,j}$$ , with $$\{v_{1}, v_{2}\} \subseteq V(F)$$ and $$i+j=d+1$$ , with a well-defined exception. Here $$B_{i, j}$$ denotes the graph obtained by attaching two vertex-disjoint paths of lengths $$i, j \ge 1$$ to a triangle. We also use the result above to strengthen the results in Xiong et al. (Discrete Math 313:784–795, 2013) in two cases, when $$i + j \le 9$$ , and when the graph is $$\Gamma _{0}$$ -free. Here $$\Gamma _{0}$$ is the simple graph with degree sequence 4, 2, 2, 2, 2. Let $$i, j > 0$$ be integers such that $$i + j \le 9$$ . Then The two results above are all sharp in the sense that the condition “ $$i+j\le 9$$ ” couldn’t be replaced by $$``i+j\le 10$$ ”.

On 1-factors with prescribed lengths in tournaments

We prove that every strongly 1050t-connected tournament contains all possible 1-factors with at most t components and this is best possible up to constant. In addition, we can ensure that each cycle in the 1-factor contains a prescribed vertex. This answers a question by Kühn, Osthus, and Townsend.Indeed, we prove more results on partitioning tournaments. We prove that a strongly Ω(k4tq)-connected tournament admits a vertex partition into t strongly k-connected tournaments with prescribed sizes such that each tournament contains q prescribed vertices, provided that the prescribed sizes are Ω(n). This result improves the earlier result of Kühn, Osthus, and Townsend. We also prove that for a strongly Ω(t)-connected n-vertex tournament T and given 2t distinct vertices x1,…,xt,y1,…,yt of T, we can find t vertex disjoint paths P1,…,Pt such that each path Pi connecting xi and yi has the prescribed length, provided that the prescribed lengths are Ω(n). For both results, the condition of connectivity being linear in t is best possible, and the condition of prescribed sizes being Ω(n) is also best possible.

The t/k-diagnosability and strong Menger connectivity on star graphs with conditional faults

The t/k-diagnosis, a famous diagnosis strategy, is proposed by Somani and Peleg. In this strategy, all the faulty vertices, no more than t, can be isolated into a faulty set, which may contain no more than k fault-free vertices. Somani and Peleg ([16], 1996) first proved that Sn was [(k+1)n−3k−2]/k-diagnosable for 0<k≤n. Chen and Liu ([4], 2012) found that the proof of that proposition was flawed and they pointed out that Sn was actually [(k+1)n−3k−1]/k-diagnosable for small k without a specific rang of k. Zhou et al. ([21], 2015) proved that Sn(n≥5) was [(k+1)n−3k−1]/k-diagnosable for 1≤k≤3 and they proposed a problem: was Sn still [(k+1)n−3k−1]/k-diagnosable for k≥4? In this paper, we prove that Sn(n≥5) is (5n−14)/4-diagnosable and it is not (5n−13)/4-diagnosable, which gives a negative answer to the problem of Zhou et al. Furthermore, we show that Sn is still strongly Menger connected even if (4n−13) vertices fail, which means there are min⁡{degG−F⁡(x),degG−F⁡(y)}-internally vertex disjoint paths between any two different vertices x,y in G−F for F∈V(Sn), δ(G−F)≥2 and |F|≤4n−13, and the bound 4n−13 is sharp.

Vertex-disjoint paths joining adjacent vertices in faulty hypercubes

Let Qn denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in Qn be denoted by Fe and Fv, respectively. In this paper, we investigate Qn (n≥3) with |Fe|+|Fv|≤n−3 faulty elements, and demonstrate that there are two fault-free vertex-disjoint paths P[a,b] and P[c,d] satisfying that 2≤ℓ(P[a,b])+ℓ(P[c,d])≤2n−2|Fv|−2, where 2|(ℓ(P[a,b])+ℓ(P[c,d])), (a,b),(c,d)∈E(Qn). The contribution of this paper is: (1) we can quickly obtain the interesting result that Qn−Fe is bipancyclic, where |Fe|≤n−2 and n≥3; (2) this result is a complement to Chen's part result (Chen (2009) [2]) in that our result shows that there are all kinds of two disjoint-free (S,T)-paths which contain 4,6,8,…,2n−2|Fv| vertices respectively in Qn when S={a,c}, T={b,d}, and (a,b),(c,d)∈E(Qn). Our result is optimal with respect to the number of fault-tolerant elements.

Berge’s Conjecture and Aharoni–Hartman–Hoffman’s Conjecture for Locally In-Semicomplete Digraphs

Let k be a positive integer and let D be a digraph. A path partition $$\mathcal {P}$$ of D is a set of vertex-disjoint paths which covers V(D). Its k-norm is defined as $$\sum _{P \in \mathcal {P}} \min \{|V(P)|, k\}$$ . A path partition is k-optimal if its k-norm is minimum among all path partitions of D. A partialk-coloring is a collection of k disjoint stable sets. A partial k-coloring $$\mathcal {C}$$ is orthogonal to a path partition $$\mathcal {P}$$ if each path $$P \in \mathcal {P}$$ meets $$\min \{|V(P)|,k\}$$ distinct sets of $$\mathcal {C}$$ . Berge (Eur J Comb 3(2):97–101, 1982) conjectured that every k-optimal path partition of D has a partial k-coloring orthogonal to it. A (path) k-pack of D is a collection of at most k vertex-disjoint paths in D. Its weight is the number of vertices it covers. A k-pack is optimal if its weight is maximum among all k-packs of D. A coloring of D is a partition of V(D) into stable sets. A k-pack $$\mathcal {P}$$ is orthogonal to a coloring $$\mathcal {C}$$ if each set $$C \in \mathcal {C}$$ meets $$\min \{|C|, k\}$$ paths of $$\mathcal {P}$$ . Aharoni et al. (Pac J Math 2(118):249–259, 1985) conjectured that every optimal k-pack of D has a coloring orthogonal to it. A digraph D is semicomplete if every pair of distinct vertices of D are adjacent. A digraph D is locally in-semicomplete if, for every vertex $$v \in V(D)$$ , the in-neighborhood of v induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge’s and Aharoni–Hartman–Hoffman’s Conjectures for locally in/out-semicomplete digraphs.

Path matrix and path energy of graphs

Given a graph G, we associate to it a path matrix P whose (i, j) entry represents the maximum number of vertex disjoint paths between the vertices i and j, with zeros on the main diagonal. In this note, we resolve four conjectures from Shikare et al. (2018) on the path energy of graphs and finally present efficient O(|E||V|3) algorithm for computing the path matrix used for verifying computational results.

One-to-one disjoint path covers in hypercubes with faulty edges

A one-to-one k-disjoint path cover $$\{P_1,P_2,\ldots ,P_k\}$$ of a graph G is a collection of k internally vertex disjoint paths joining source with sink that cover all vertices of G. In this paper, we investigate the problem of one-to-one disjoint path cover in hypercubes with faulty edges and obtain the following results: Let u, v ∈ V(Qn) be such that $$p(u)\ne p(v)$$ and $$1\le k\le n$$ . Then there exists a one-to-one k-disjoint path cover $$\{P_1,P_2,\ldots ,P_k\}$$ joining vertices u and v in $$Q_n$$ . Moreover, when $$1\le k\le n-2$$ , the result still holds even if removing $$n-2-k$$ edges from $$Q_n$$ .

Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes

The n-dimensional balanced hypercube BHn (n≥1) has been proved to be a bipartite graph. Let P be a set of edges in BHn. For any two vertices u,v from different partite sets of V(BHn). In this paper, we prove that if |P|≤2n−2 and the subgraph induced by P has neither u nor v as internal vertices , or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. As a corollary, if |P|≤2n−1, then BHn contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.

Disjoint path covers joining prescribed source and sink sets in interval graphs

A disjoint path cover of a graph is a set of internally vertex-disjoint paths that altogether cover every vertex of the graph. Given two disjoint source and sink sets, S and T, in a graph G, a k-disjoint path cover of G joining S and T is a disjoint path cover composed of k paths, each of which runs from a source in S to a sink in T. In this paper, we give two short proofs for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-one type) joining S and T for any S and T with |S|=|T|=1 and for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-many type) joining S and T for any S and T with |S|=1 and |T|=k. Also, some partial results on the many-to-many disjoint path coverability of an interval graph are addressed. In addition, we discuss interval graphs in which the disjoint-path-cover property is preserved after removal of arbitrary f or less vertices.

Matching, path covers, and total forcing sets

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The minimum cardinality of a total forcing set in G is its total forcing number, denoted Ft(G). The path cover number of G, denoted pc(G), is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover, while the matching number of G, denoted α′(G), is the number of edges in a maximum matching of G. Let T be a tree of order at least two. We observe that pc(T) + 1 ≤ Ft(T) ≤ 2pc(T), and we prove that Ft(T) ≤ α′(T) + pc(T). Further, we characterize the extremal trees achieving equality in these bounds.

An improved approximation algorithm for the minimum 3-path partition problem

Given a graph $$G = (V, E)$$ , we seek for a collection of vertex disjoint paths each of order at most 3 that together cover all the vertices of V. The problem is called 3-path partition, and it has close relationships to the well-known path cover problem and the set cover problem. The general k-path partition problem for a constant $$k \ge 3$$ is NP-hard, and it admits a trivial k-approximation. When $$k = 3$$ , the previous best approximation ratio is 1.5 due to Monnot and Toulouse (Oper Res Lett 35:677–684, 2007), based on two maximum matchings. In this paper we first show how to compute in polynomial time a 3-path partition with the least 1-paths, and then apply a greedy approach to merge three 2-paths into two 3-paths whenever possible. Through an amortized analysis, we prove that the proposed algorithm is a 13 / 9-approximation. We also show that the performance ratio 13 / 9 is tight for our algorithm.