Local edge and local node connectivity in regular graphs

2003 ◽  
Vol 40 (1-2) ◽  
pp. 151-158
Author(s):  
O. Fülöp
Keyword(s):  
Undirected Graph ◽  
Simple Graph ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Connectivity ◽  
Edge Disjoint ◽  
Node Connectivity ◽  
Multiple Edges ◽  
Local Edge

W. Mader [5] proved that every undirected graph (multiple edges are allowed but loops not) contains adjacent nodes x and y joined by min (d(x),dG(y))G edge-disjoint paths and in every undirected simple graph there are two adjacent nodes x and y joined by min (d(x),dG(y) Ginternally node-disjoint paths. In general it is not possible to fix x (or y) arbitrarily. The purpose of this paper is to provide conditions for the existence of a node x in d-regular graphs such that for all y joined to x there are d pairwise edge-disjoint (node-disjoint) paths between x and y. We also examine the directed version in case of local edge connectivity.

scholarly journals Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

2000 ◽  
Vol 9 (3) ◽  
pp. 241-263 ◽  
Author(s):  
ALAN M. FRIEZE ◽  
LEI ZHAO
Keyword(s):  
Polynomial Time ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Constant Factor ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Disjoint ◽  
Random Regular Graph ◽  
Arbitrary Graphs ◽  
Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

scholarly journals Double Arithmetic Odd Decomposition [DAOD] of Some Complete 4-Partite Graphs

2019 ◽  
Vol 9 (2) ◽  
pp. 3902-3907
Keyword(s):  
Undirected Graph ◽  
Edge Disjoint ◽  
Multiple Edges

Let G be a finite, connected, undirected graph without loops or multiple edges. If G1 , G2 , . . . ,Gn are connected edge – disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  . . .  E(Gn), then { G1 , G2 , . . . , Gn} is said to be a decomposition of G. The concept of Arithmetic Odd Decomposition [AOD] was introduced by E. Ebin Raja Merly and N. Gnanadhas . A decomposition {G1 , G2 , . . . , Gn } G is said to be Arithmetic Decomposition if each Gi is connected and | E(Gi )| = a+ (i – 1) d , for 1  i  n and a, d  ℤ . When a =1 and d = 2, we call the Arithmetic Decomposition as Arithmetic Odd Decomposition . A decomposition { G1 , G3 , . . . , G2n-1} of G is said to be AOD if | E (Gi ) | = i ,  i = 1, 3, . . . , 2n-1. In this paper, we introduce a new concept called Double Arithmetic Odd Decomposition [DAOD]. A graph G is said to have Double Arithmetic Odd Decomposition [DAOD] if G can be decomposed into 2k subgraphs { 2G1 , 2G3 , . . . , 2G2k-1 } such that each Gi is connected and | E (Gi ) | = i ,  i = 1, 3, . . . , 2k-1. Also we investigate DAOD of some complete 4-partite graphs such as K2,2,2,m , K2,4,4,m and K1 ,2,4,m .

Average Edge-Connectivity of Cubic Graphs

2021 ◽  
pp. 2142002
Author(s):  
Miaomiao Zhuo ◽  
Qinqin Li ◽  
Baoyindureng Wu ◽  
Xinhui An
Keyword(s):  
Disjoint Paths ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Edge Connectivity ◽  
Edge Disjoint

In this paper, we consider the concept of the average edge-connectivity [Formula: see text] of a graph [Formula: see text], defined to be the average, over all pairs of vertices, of the maximum number of edge-disjoint paths connecting these vertices. Kim and O previously proved that [Formula: see text] for any connected cubic graph on [Formula: see text] vertices. We refine their result by showing that [Formula: see text] We also characterize the graphs where equality holds.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

2021 ◽  
pp. 1-11
Author(s):  
Pingshan Li ◽  
Rong Liu ◽  
Xianglin Liu
Keyword(s):  
Cayley Graph ◽  
Fault Tolerant ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Star Graph ◽  
Edge Connectivity ◽  
Short Edge ◽  
Edge Disjoint

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.

scholarly journals Strong Menger Connectedness of Augmented k-ary n-cubes

2020 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Arbitrary Vertex ◽  
Edge Disjoint ◽  
Vertex Set ◽  
Delta G ◽  
Disjoint Edge

Abstract A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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