scholarly journals Super Connectivity of Erdős-Rényi Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 267 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graphs ◽  
Minimum Cardinality ◽  
Super Connectivity ◽  
Isolated Vertex ◽  
Disconnected Graph

The super connectivity κ ′ ( G ) of a graph G is the minimum cardinality of vertices, if any, whose deletion results in a disconnected graph that contains no isolated vertex. G is said to be r-super connected if κ ′ ( G ) ≥ r . In this note, we establish some asymptotic almost sure results on r-super connectedness for classical Erdős–Rényi random graphs as the number of nodes tends to infinity. The known results for r-connectedness are extended to r-super connectedness by pairing off vertices and estimating the probability of disconnecting the graph that one gets by identifying the two vertices of each pair.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

scholarly journals The Unit Acquisition Number of Binomial Random Graphs

10.37236/9671 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Konstantinos Georgiou ◽  
Somnath Kundu ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Connected Graph ◽  
Unit Weight ◽  
Time Step ◽  
Minimum Cardinality ◽  
Positive Weight

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The unit acquisition number of $G$, denoted by $a_u(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdős-Rényi random graph process $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choose 2}$. We show that asymptotically almost surely $a_u(\mathcal{G}(n,m)) = 1$ right at the time step the random graph process creates a connected graph. Since trivially $a_u(\mathcal{G}(n,m)) \ge 2$ if the graphs is disconnected, the result holds in the strongest possible sense.

Restrained geodetic domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050084
Author(s):  
John Joy Mulloor ◽  
V. Sangeetha
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Geodetic Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Domination In Graphs ◽  
Corona Product

Let [Formula: see text] be a graph with edge set [Formula: see text] and vertex set [Formula: see text]. For a connected graph [Formula: see text], a vertex set [Formula: see text] of [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] lies in a shortest path between any pair of vertices in [Formula: see text]. If the geodetic set [Formula: see text] is dominating, then [Formula: see text] is geodetic dominating set. A vertex set [Formula: see text] of [Formula: see text] is said to be a restrained geodetic dominating set if [Formula: see text] is geodetic, dominating and the subgraph induced by [Formula: see text] has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.

Vertex criticality with respect to isolate domination

2015 ◽  
Vol 07 (02) ◽  
pp. 1550010 ◽  
Author(s):  
I. Sahul Hamid ◽  
S. Balamurugan
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Extended Chain ◽  
Isolated Vertex ◽  
Domination In Graphs

A set S of vertices of a graph G is called a dominating set of G if every vertex in V(G) - S is adjacent to a vertex in S. A dominating set S such that the subgraph 〈S〉 induced by S has at least one isolated vertex is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate domination number and is denoted by γ0(G). This concept was introduced in [I. Sahul Hamid and S. Balamurugan, Isolate domination in graphs (Communicated)] and further studied in [I. Sahul Hamid and S. Balamurugan, Extended chain of domination parameters in graphs, ISRN Combin. 2013 (2013), Article ID: 792743, 4 pp.; Isolate domination and maximum degree, Bull. Int. Math. Virtual Inst. 3 (2013) 127–133; Isolate domination in unicyclic graphs, Int. J. Math. Soft Comput. 3(3) (2013) 79–83]. This paper studies the effect of the removal of a vertex upon the isolate domination number.

scholarly journals Bounding the paired-domination number of a tree in terms of its annihilation number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 523-529 ◽  
Author(s):  
Nasrin Dehgardi ◽  
Seyed Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Degree Sequence ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Isolated Vertex ◽  
Paired Domination

A paired-dominating set of a graph G=(V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by ?pr(G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n?2,?pr(T)? 4a(T)+2/3 and we characterize the trees achieving this bound.

Non-Isolated Resolving Number of Graph with Pendant Edges

2019 ◽  
Vol 19 (02) ◽  
pp. 1950003
Author(s):  
RIDHO ALFARISI ◽  
DAFIK ◽  
ARIKA INDAH KRISTIANA ◽  
IKA HESTI AGUSTIN
Keyword(s):  
Connected Graph ◽  
Ordered Set ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Isolated Vertex

We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, …, wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges.

Doubly connected bi-domination in graphs

2020 ◽  
pp. 2150009 ◽  
Author(s):  
Mohammed A. Abdlhusein
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Finite Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Isolated Vertex ◽  
Domination In Graphs

Let [Formula: see text] be a finite graph, simple, undirected and has no isolated vertex. A dominating subset [Formula: see text] of [Formula: see text] is said a bi-dominating set, if every vertex of it dominates two vertices of [Formula: see text]. The bi-domination number of [Formula: see text], denoted by [Formula: see text] is the minimum cardinality over all bi-dominating sets in [Formula: see text]. In this paper, a certain modified bi-domination parameter called doubly connected bi-domination and its inverse are introduced. Several bounds and properties are studied here. These modified dominations are applied and evaluated for several well-known graphs and complement graphs.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

The total Steiner number of a graph

2020 ◽  
Vol 12 (03) ◽  
pp. 2050038
Author(s):  
J. John
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
General Properties ◽  
Isolated Vertex

A total Steiner set of [Formula: see text] is a Steiner set [Formula: see text] such that the subgraph [Formula: see text] induced by [Formula: see text] has no isolated vertex. The minimum cardinality of a total Steiner set of [Formula: see text] is the total Steiner number of [Formula: see text] and is denoted by [Formula: see text]. Some general properties satisfied by this concept are studied. Connected graphs of order [Formula: see text] with total Steiner number 2 or 3 are characterized. We partially characterized classes of graphs of order [Formula: see text] with total Steiner number equal to [Formula: see text] or [Formula: see text] or [Formula: see text]. It is shown that [Formula: see text]. It is shown that for every pair k, p of integers with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text]. Also, it is shown that for every positive integer [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text] and [Formula: see text].

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