A -approximation for the minimum 2-local-vertex-connectivity augmentation in a connected graph

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components and a minimum vertex separator is a vertex separator of least cardinality. The vertex connectivity refers to the size of a minimum vertex separator. For a connected graph G with vertex connectivity [Formula: see text], the connectivity augmentation refers to a set S of edges whose augmentation to G increases its vertex connectivity by one. A minimum connectivity augmentation of G is the one in which S is minimum. In this paper, we focus our attention on biconnectivity augmentation for trees. Towards this end, we present a new sequential algorithm for biconnectivity augmentation in trees by simplifying the algorithm reported in [1]. The simplicity is achieved with the help of edge contraction tool. This tool helps us in getting a recursive subproblem preserving all connectivity information. Subsequently, we present a parallel algorithm to obtain a minimum biconnectivity augmentation set in trees. Our parallel algorithm essentially follows the overall structure of sequential algorithm. Our implementation is based on CREW PRAM model with [Formula: see text] processors, where [Formula: see text] refers to the maximum degree of a tree. We also show that our parallel algorithm is optimal with a processor-time product of [Formula: see text] where n is the number of vertices of a tree.

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A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs

The Computer Journal ◽

10.1093/comjnl/bxz077 ◽

2019 ◽

Vol 63 (9) ◽

pp. 1372-1384

Author(s):

Zuwen Luo ◽

Liqiong Xu

Keyword(s):

Fault Tolerance ◽

Connected Graph ◽

Cayley Graphs ◽

Network Structures ◽

Vertex Connectivity ◽

Wheel Graphs

Abstract Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$-vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$. The cardinality of a minimum $R^{k}$-vertex-cut is the $R^{k}$-vertex-connectivity of $G$ and is denoted by $\kappa ^{k}(G)$. $R^{k}$-vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$-vertex-connectivity and $R^{2}$-vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $\kappa ^{1}(AW_{n})=4n-7$ for $n\geq 6$; $\kappa ^{2}(AW_{n})=6n-12$ for $n\geq 6$.

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k-Edge and 3-Vertex Connectivity Augmentation in an Arbitrary Multigraph

Algorithms and Computation - Lecture Notes in Computer Science ◽

10.1007/3-540-49381-6_18 ◽

1998 ◽

pp. 159-169 ◽

Cited By ~ 4

Author(s):

Toshimasa Ishii ◽

Hiroshi Nagamochi ◽

Toshihide Ibaraki

Keyword(s):

Vertex Connectivity ◽

Connectivity Augmentation

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On the distance signless Laplacian spectral radius of graphs and digraphs

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.1982 ◽

2017 ◽

Vol 32 ◽

pp. 438-446 ◽

Cited By ~ 6

Author(s):

Dan Li ◽

Guoping Wang ◽

Jixiang Meng

Keyword(s):

Spectral Radius ◽

Connected Graph ◽

Minimum Degree ◽

Extremal Graph ◽

Minimal Distance ◽

Vertex Connectivity ◽

Laplacian Spectral Radius ◽

Connected Graphs ◽

Signless Laplacian Spectral Radius ◽

Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

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Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

Approximation and Online Algorithms - Lecture Notes in Computer Science ◽

10.1007/978-3-030-92702-8_1 ◽

2021 ◽

pp. 1-22

Author(s):

Waldo Gálvez ◽

Francisco Sanhueza-Matamala ◽

José A. Soto

Keyword(s):

Approximation Algorithms ◽

Vertex Connectivity ◽

Connectivity Augmentation

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General eccentric distance sum of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500464 ◽

2020 ◽

pp. 2150046

Author(s):

Tomáš Vetrík

Keyword(s):

Connected Graph ◽

Bipartite Graphs ◽

Extremal Graphs ◽

Vertex Connectivity ◽

Vertex Set ◽

General Graphs ◽

Eccentric Distance

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

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