Exponential Independence Number of Some Graphs

2018 ◽  
Vol 29 (07) ◽  
pp. 1151-1164
Author(s):  
Canan Çiftçi ◽  
Aysun Aytaç
Keyword(s):  
Shortest Path ◽  
Independence Number ◽  
General Theorem ◽  
Independent Set ◽  
Maximum Order ◽  
Tree Graphs ◽  
Number Formula ◽  
Thorn Graph

Let [Formula: see text] be a graph and [Formula: see text]. We define by [Formula: see text] the subgraph of [Formula: see text] induced by [Formula: see text]. For each vertex [Formula: see text] and for each vertex [Formula: see text], [Formula: see text] is the length of the shortest path in [Formula: see text] between [Formula: see text] and [Formula: see text] if such a path exists, and [Formula: see text] otherwise. For a vertex [Formula: see text], let [Formula: see text] where [Formula: see text]. Jäger and Rautenbach [27] define a set [Formula: see text] of vertices to be exponential independent if [Formula: see text] for every vertex [Formula: see text] in [Formula: see text]. The exponential independence number [Formula: see text] of [Formula: see text] is the maximum order of an exponential independent set. In this paper, we give a general theorem and we examine exponential independence number of some tree graphs and thorn graph of some graphs.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

Strong vb-dominating and vb-independent sets of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050002 ◽  
Author(s):  
Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Number Formula

Let [Formula: see text] be a graph. A vertex [Formula: see text] strongly (weakly) b-dominates block [Formula: see text] if [Formula: see text] ([Formula: see text]) for every vertex [Formula: see text] in the block [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in [Formula: see text] is strongly (weakly) b-dominated by some vertex in [Formula: see text]. The strong (weak) vb-domination number [Formula: see text] ([Formula: see text]) is the order of a minimum SVBD (WVBD) set of [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if [Formula: see text] is a vertex block independent set and for every vertex [Formula: see text] and every block [Formula: see text] incident on [Formula: see text], there exists a vertex [Formula: see text] in the block [Formula: see text] such that [Formula: see text] ([Formula: see text]). The strong (weak) vb-independence number [Formula: see text] ([Formula: see text]) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of [Formula: see text]. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.

scholarly journals On the k -Component Independence Number of a Tree

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Shuting Cheng ◽  
Baoyindureng Wu
Keyword(s):  
Linear Time ◽  
Independence Number ◽  
Independent Set ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Maximum Order ◽  
Component Order

Let G be a graph and k ≥ 1 be an integer. A subset S of vertices in a graph G is called a k -component independent set of G if each component of G S has order at most k . The k -component independence number, denoted by α c k G , is the maximum order of a vertex subset that induces a subgraph with maximum component order at most k . We prove that if a tree T is of order n , then α k T ≥ k / k + 1 n . The bound is sharp. In addition, we give a linear-time algorithm for finding a maximum k -component independent set of a tree.

Turán-Ramsey Theorems and Kp-Independence Numbers

1994 ◽  
Vol 3 (3) ◽  
pp. 297-325 ◽  
Author(s):  
P. Erdős ◽  
A. Hajnal ◽  
M. Simonovits ◽  
V. T. Sós ◽  
E. Szemerédi
Keyword(s):  
Simple Structure ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Asymptotic Value ◽  
Image Position ◽  
Independence Numbers ◽  
Edge Colouring

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value offor p < q. Several results are proved, and some other problems and conjectures are stated.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

VERTEX VULNERABILITY PARAMETER OF GEAR GRAPHS

2011 ◽  
Vol 22 (05) ◽  
pp. 1187-1195 ◽  
Author(s):  
AYSUN AYTAC ◽  
TUFAN TURACI
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Minimum Cardinality ◽  
Independent Domination

For a vertex v of a graph G = (V,E), the independent domination number (also called the lower independence number) iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is [Formula: see text]. In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offered.

Trees with independent Roman domination number twice the independent domination number

2015 ◽  
Vol 07 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Mustapha Chellali ◽  
Nader Jafari Rad
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Open Question ◽  
Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

Outer-connected independence in graphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550039
Author(s):  
I. Sahul Hamid ◽  
R. Gnanaprakasam ◽  
M. Fatima Mary
Keyword(s):  
Independence Number ◽  
Independent Set ◽  
Proper Subset ◽  
Maximum Cardinality

A set S ⊆ V(G) is an independent set if no two vertices of S are adjacent. An independent set S such that 〈V - S〉 is connected is called an outer-connected independent set(oci-set). An oci-set is maximal if it is not a proper subset of any oci-set. The minimum and maximum cardinality of a maximal oci-set are called respectively the outer-connected independence number and the upper outer-connected independence number. This paper initiates a study of these parameters.

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