Paths P1 Research Articles

The 3-path-connectivity of the star graphs

Let G be a graph with vertex set V(G) and S⊆V(G). An S-path of G is a path which connects all vertices of S in G. Let P1 and P2 be S-paths. If E(P1)∩E(P2)=0̸ and V(P1)∩V(P2)=S, then the two paths P1 and P2 are called internally disjointS-paths. Denoted by πG(S) the maximum number of any pairwise internally disjoint S-paths in G. The r-path-connectivityπr(G) is defined as πr(G)= min{πG(S)|S⊆V(G),|S|=r}, where 2≤r≤|V(G)|. In this paper, we focus on the 3-path-connectivity of the n-dimensional star graph Sn and obtain that π3(Sn)=3(n−1)−14 with n≥3.

A graph minor condition for graphs to be [formula omitted]-linked

A graph is called k-linked, if for any 2k distinct vertices x1,x2,…,xk,y1,y2,…,yk, there exist k vertex disjoint paths P1,P2,…,Pk such that Pi connects xi and yi for each 1≤i≤k. Robertson and Seymour showed that every 2k-connected graph having K3k as a minor is k-linked. In 2005, Chen, Gould, Kawarabayashi, Pfender, and Wei proved that every 6-connected graph having K9− as a minor is 3-linked, where Kk−i is the graph obtained from the complete graph with k vertices by deleting exactly i edges. We improve these two results by showing that every 2k-connected graph having the graph obtained from K3k by deleting independent k edges as a minor is k-linked.

Improved results on linkage problems

A digraph D is k-linked if |D|≥2k, and whenever x1,…,xk,y1,…,yk are 2k distinct vertices of D, there exist vertex-disjoint paths P1,…,Pk such that Pi is a path from xi to yi for each i∈[k]. A digraph is strongly connected if it has a directed path from x to y for every ordered pair of distinct vertices x,y and it is strongly k-connected (briefly k-connected) if it has at least k+1 vertices and remains strongly connected when we delete any set of at most k−1 vertices. In this paper, we prove that every (13k−6)-connected semicomplete digraph with minimum out-degree at least 19k−6 is k-linked. This result extends the result of Bang-Jensen and Johansen (2022) [3] to semicomplete digraphs. Moreover, we show the following holds. Let n,k,l∈N, and let D be a digraph of order n. Suppose that D is (39k+9l−28)-connected with minimum degree at least n−l and minimum out-degree at least 46k+15l−35. Then D is k-linked.

Few induced disjoint paths for H-free graphs

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

Every (13k − 6)-strong tournament with minimum out-degree at least 28k − 13 is k-linked

A digraph D is k-linked if it satisfies that for every choice of disjoint sets {x1,…,xk} and {y1,…,yk} of vertices of D there are vertex disjoint paths P1,…,Pk such that Pi is an (xi,yi)-path. Confirming a conjecture by Kühn et al., Pokrovskiy proved in 2015 that every 452k-strong tournament is k-linked and asked for a better linear bound. Very recently Meng et al. proved that every (40k−31)-strong tournament is k-linked. In this note we use an important lemma from their paper to give a short proof that every (13k−6)-strong tournament of minimum out-degree at least 28k−13 is k-linked.

Induced Disjoint Paths in AT-free graphs

Paths P1,…,Pk are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This problem is NP-complete even for k=2. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. Consequently, the problem of deciding if an AT-free graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. We show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH|, even for a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-in-a-Path and k-in-a-Tree are polynomial-time solvable on AT-free graphs even if k is part of the input.

An improved linear connectivity bound for tournaments to be highly linked

A digraph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi is directed from xi to yi for i=1,…,k. Pokrovskiy in 2015 proved that every strongly 452k-connected tournament is k-linked. In this paper, we significantly reduce this connectivity bound and show that any (40k−31)-connected tournament is k-linked.

Rooted topological minors on four vertices

For a graph G and a set Z of four distinct vertices of G, a diamond on Z is a subgraph of G such that, for some labeling Z={v1,v2,v3,v4}, there are three internally disjoint paths P1,P2,P3 with end vertices v1,v2 with v3,v4 on P1,P2, respectively. Therefore, this yields a K4−-subdivision with branch vertices on Z.We characterize graphs G that contain no diamond on a prescribed set Z of four vertices, under the assumption that for every v∈Z there are three paths of G from v to Z−{v}, mutually disjoint except for v. Moreover, we can find two “different” such subdivisions, if one exists.Our proof is based on Mader's S-paths theorem.

The directed 2-linkage problem with length constraints

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1,s2,t1,t2 whether D contains a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8]. Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k1,k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path of length no more than d(si,ti)+ki, for i=1,2, where d(si,ti) denotes the length of the shortest (si,ti)-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1,P2 to the weak 2-linkage problem where each path Pi has length at most d(si,ti)+clog1+ϵ⁡n for some constant c.

On multi-state two separate minimal paths reliability problem with time and budget constraints

In a stochastic-flow network, a (d, T, b, P1, P2)-MP is a system state vector for which d units of flow can be transmitted through two separate minimal paths (SMPs) P1 and P2 simultaneously from a source node to a sink node satisfying time and budget limitations T and b, respectively. Problem of determining all the (d, T, b, P1, P2)-MPs, termed as the (d, T, b, P1, P2)-MP problem, has been attractive in reliability theory. Here, some new results are established for the problem. Using these results, a new algorithm is developed to find all the (d, T, b, P1, P2)-MPs, and its correctness is established. The algorithm is compared with a recently proposed one to show the practical efficiency of the algorithm.

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.

Two disjoint shortest paths problem with non-negative edge length

In the two disjoint shortest paths problem ( 2-DSPP), the input is a graph (or a digraph) and its vertex pairs (s1,t1) and (s2,t2), and the objective is to find two vertex-disjoint paths P1 and P2 such that Pi is a shortest path from si to ti for i=1,2, if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function.

Induced disjoint paths in circular-arc graphs in linear time

The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤i<j≤k. We present a linear-time algorithm that solves Induced Disjoint Paths and finds the corresponding paths (if they exist) on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (si,ti) are not necessarily distinct. In both cases, if a representation of the graph is given, then the algorithms run in O(n+k) time.

Highly linked tournaments

A (possibly directed) graph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi goes from xi to yi. A theorem of Bollobás and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás–Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments—there is a function f(k) such that every strongly f(k)-connected tournament is k-linked. The bound on f(k) was reduced to O(klog⁡k) by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked.

Odd - Even Graceful Labeling for Different Paths using Padavon Sequence

A function f is called an odd-even graceful labeling of a graph G if f: V(G) → {0,1,2,...,q} is injective and the induced function f : E(G) → { { 0,2,4,...,2q+2i/i= 1 to n} such that when each edge uv is assigned the label |f(u) – f(v)| the resulting edge labels are {2,4,6,...,2q}. A graph which admits an odd-even graceful labeling is called an odd-even graceful graph. In this paper, the odd-even gracefulness of paths p1, p2, p3,..., p11 is obtained.

Edge Odd Graceful Labeling for Different Paths using Padavon Sequence

The even graceful labeling of a graph G with q edges means that there is an injection f : V(G) to { 0,1,1,1,2,...,2q+i/i= 1 to n} such that when each edge uv is assigned the label |f(u) – f(v)| the resulting edge labels are {1,3,5,...,2q-1}. A graph which admits an edge odd graceful labeling is called an edgeodd graceful graph. In this paper, the edge odd gracefulness of paths p1, p2, p3,..., p12 is obtained.

Two spanning disjoint paths with required length in generalized hypercubes

Given two pairs 〈u, v〉 and 〈x, y〉 of vertices of a graph G=(V,E) and two integers l1 and l2 with l1+l2=|V(G)|−2, G is said to be satisfying the 2RP-property if there exist two disjoint paths P1 and P2 such that (1) P1 is a path joining u to v with l(P1)=l1, (2) P2 is a path joining x to y with l(P2)=l2, and (3) P1∪P2 spans G, where l(P) denotes the length of path P. In this paper, we show that an r-dimensional generalized hypercube, denoted by G(mr,mr−1,…,m1), satisfies the 2RP-property except some special conditions, where mi⩾4 for all 1⩽i⩽r.

Paired many-to-many disjoint path covers of the hypercubes

In this paper we consider the problem of paired many-to-many disjoint path covers of the hypercubes and obtain the following result. Let S={s1,s2,…,sk} and T={t1,t2,…,tk} be two sets of k vertices in different partite sets of the n-dimensional hypercube Qn, and let e=|{i|siandtiare adjacent,1⩽i⩽k}|. If n>k+⌈(k-e)/2⌉, then there exist k vertex-disjoint paths P1,P2,…,Pk, where Pi connects si and ti, for i=1,2,…,k, such that these k paths contain all vertices of Qn.

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.

An Improved Algorithm for the Half-Disjoint Paths Problem

In this paper, we consider the half-integral disjoint paths packing. For a graph G and k pairs of vertices (s1,t1),(s2,t2),…,(sk,tk) in G, the objective is to find paths P1,…,Pk in G such that Pi joins si and ti for i=1,2,…,k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k=O((logn/loglogn)1/7). This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530–539] who proved the same conclusion when k=O((loglogn)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a “crossbar” of a polynomial size of the tree width of the graph.