scholarly journals REPARTITORS, SELECTORS AND SUPERSELECTORS

2006 ◽  
Vol 07 (03) ◽  
pp. 391-415 ◽  
Author(s):  
FRÉDÉRIC HAVET
Keyword(s):  
Lower Bounds ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Vertex Set ◽  
Optimal Networks

An (n, p, f)-network G is a graph (V, E) where the vertex set V is partitioned into four subsets [Formula: see text] and [Formula: see text] called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are p priorities, n - p ordinary inputs and n + f outputs; each priority, each ordinary input and each output is connected to exactly one switch; switches have degree at most 4. An (n, p, f)-network is an (n, p, f)-repartitor if for any disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist in G, n edge-disjoint paths, p of them from [Formula: see text] to [Formula: see text] and the n - p others joining [Formula: see text] to [Formula: see text]. The problem is to determine the minimum number R(n, p, f) of switches of an (n, p, f)-repartitor and to construct a repartitor with the smallest number of switches. In this paper, we show how to build general repartitors from (n, 0, f)-repartitors also called (n, n + f)-selectors. We then consrtuct selectors using more powerful networks called superselectors. An (n, 0, 0)-network is an n-superselector if for any subsets [Formula: see text] and [Formula: see text] with [Formula: see text], there exist in G, [Formula: see text] edge-disjoint paths joining [Formula: see text] to [Formula: see text]. We show that the minimum number of switches of an n-superselector S+ (n) is at most 17n + O(log(n)). We then deduce that [Formula: see text] if [Formula: see text], R(n, p, f) ≤ 18n + 34f + O( log (n + f)), if [Formula: see text] and [Formula: see text] if [Formula: see text]. Finally, we give lower bounds for R(n, 0, f) and S+ (n) and show optimal networks for small value of n.

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.

scholarly journals A study on the pendant number of graph products

2019 ◽  
Vol 11 (1) ◽  
pp. 24-40
Author(s):  
Jomon K. Sebastian ◽  
Joseph Varghese Kureethara ◽  
Sudev Naduvath ◽  
Charles Dominic
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Disjoint Paths ◽  
Graph Products ◽  
Path Decomposition ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Number Π

Abstract A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

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 Two coloring problems on matrix graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650053
Author(s):  
Zhe Han ◽  
Mei Lu
Keyword(s):  
Finite Field ◽  
Optical Networks ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Basic Properties ◽  
New Family ◽  
Minimum Number ◽  
Vertex Set ◽  
Positive Integers

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set [Formula: see text] is the set of all [Formula: see text] matrices over a finite field [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text]. And any two matrices share an edge if the rank of their difference is [Formula: see text]. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let [Formula: see text] (resp., [Formula: see text]) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most [Formula: see text] (resp., exactly [Formula: see text]) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of [Formula: see text] and give some upper and lower bounds on [Formula: see text].

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 Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

On complete subgraphs of different orders

1976 ◽  
Vol 79 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Intersection Graph ◽  
Edge Disjoint ◽  
Vertex Set

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

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