scholarly journals On the Paired-Domination Subdivision Number of Trees

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1135
Author(s):  
Shouliu Wei ◽  
Guoliang Hao ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Hossein Karami
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Minimum Number ◽  
Paired Domination ◽  
Domination Subdivision Number

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T≠P5 of order n≥3 and each edge e∉E(T), sdγpr(T)+sdγpr(T+e)≤n+2.

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.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

scholarly journals Upper paired domination versus upper domination

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Graph Classes ◽  
Paired Domination Number ◽  
Upper Domination Number ◽  
Paired Domination ◽  
Cactus Graphs ◽  
Open Question

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals Locating Equitable Domination and Independence Subdivision Numbers of Graphs

2014 ◽  
Vol 9 ◽  
pp. 27-32 ◽  
Author(s):  
P. Sumathi ◽  
G. Alarmelumangai
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Domination Subdivision Number

Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wєV-D, N(u)∩D ≠ N(w)∩D, |N(u)∩D| ≠ |N(w)∩D|. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the locating equitable domination number and is denoted by sdγle(G). The independence subdivision number sdβle(G) to equal the minimum number of edges that must be subdivided in order to increase the independence number. In this paper, we establish bounds on sdγle(G) and sdβle(G) for some families of graphs.

scholarly journals Γ -Paired dominating graphs of some paths

2018 ◽  
Vol 189 ◽  
pp. 03029
Author(s):  
Pannawat Eakawinrujee ◽  
Nantapath Trakultraipruk
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
The Other ◽  
Maximum Cardinality ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination

A paired dominating set of a graph G = (V(G),E(G)) is a set D of vertices of G such that every vertex is adjacent to some vertex in D, and the subgraph of G induced by D contains a perfect matching. The upper paired domination number of G, denoted by Γpr(G) is the maximum cardinality of a minimal paired dominating set of G. A paired dominatin set of cardinality Γ pr(G) is called a Γpr(G) -set. The Γ -paired dominating graph of G, denoted by ΓPD(G), is the graph whose vertex set is the set of all Γ pr(G) -sets, and two Γpr(G) -sets are adjacentin ΓPD(G) if one can be obtained from the other by removing one vertex and adding another vertex of G. In this paper, we present the Γ-paired dominating graphs of some paths.

Secure domination subdivision number of a graph

2019 ◽  
Vol 11 (03) ◽  
pp. 1950036
Author(s):  
S. V. Divya Rashmi ◽  
A. Somasundaram ◽  
S. Arumugam
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Number Formula ◽  
Secure Domination ◽  
Domination Subdivision Number

Let [Formula: see text] be a graph of order [Formula: see text] and size [Formula: see text] A dominating set [Formula: see text] of [Formula: see text] is called a secure dominating set if for each [Formula: see text] there exists [Formula: see text] such that [Formula: see text] is adjacent to [Formula: see text] and [Formula: see text] is a dominating set of [Formula: see text] In this case, we say that [Formula: see text] is [Formula: see text]-defended by [Formula: see text] or [Formula: see text] [Formula: see text]-defends [Formula: see text] The secure domination number [Formula: see text] is the minimum cardinality of a secure dominating set of [Formula: see text] The secure domination subdivision number of [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the secure domination number. In this paper, we present several results on this parameter.

scholarly journals Weakly convex domination subdivision number of a graph

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2101-2110
Author(s):  
Magda Dettlaff ◽  
Saeed Kosary ◽  
Magdalena Lemańska ◽  
Seyed Sheikholeslami
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Weakly Convex ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Convex Domination Number ◽  
Domination Subdivision Number

A set X is weakly convex in G if for any two vertices a,b ? X there exists an ab-geodesic such that all of its vertices belong to X. A set X ? V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number ?wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd?wcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.

scholarly journals On the Paired-Domination Subdivision Number of a Graph

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 439
Author(s):  
Guoliang Hao ◽  
Seyed Mahmoud Sheikholeslami ◽  
Mustapha Chellali ◽  
Rana Khoeilar ◽  
Hossein Karami
Keyword(s):  
Domination Number ◽  
Free Graph ◽  
Paired Domination Number ◽  
Minimum Number ◽  
Paired Domination ◽  
Domination Subdivision Number

In order to increase the paired-domination number of a graph G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number sdγpr(G) of G. It is well known that sdγpr(G+e) can be smaller or larger than sdγpr(G) for some edge e∉E(G). In this note, we show that, if G is an isolated-free graph different from mK2, then, for every edge e∉E(G), sdγpr(G+e)≤sdγpr(G)+2Δ(G).

Domination Theory in Graphs

2020 ◽  
pp. 1-23
Author(s):  
E. Sampathkumar ◽  
L. Pushpalatha
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Correct Conclusion ◽  
Minimum Number ◽  
Domination In Graphs

The study of domination in graphs originated around 1850 with the problems of placing minimum number of queens or other chess pieces on an n x n chess board so as to cover/dominate every square. The rules of chess specify that in one move a queen can advance any number of squares horizontally, vertically, or diagonally as long as there are no other chess pieces in its way. In 1850 enthusiasts who studied the problem came to the correct conclusion that all the squares in an 8 x 8 chessboard can be dominated by five queens and five is the minimum such number. With very few exceptions (Rooks, Bishops), these problems still remain unsolved today. Let G = (V,E) be a graph. A set S ⊂ V is a dominating set of G if every vertex in V–S is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set.

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