scholarly journals A Note on the Paired-Domination Subdivision Number of Trees

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 181 ◽  
Author(s):  
Xiaoli Qiang ◽  
Saeed Kosari ◽  
Zehui Shao ◽  
Seyed Mahmoud Sheikholeslami ◽  
Mustapha Chellali ◽  
...  
Keyword(s):  
Upper Bound ◽  
Isolated Vertex ◽  
Paired Domination ◽  
Domination Subdivision Number

For a graph G with no isolated vertex, let γpr(G) and sdγpr(G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n≥4 different from a healthy spider (subdivided star), then sdγpr(T)≤min{γpr(T)2+1,n2}, improving the (n−1)-upper bound that was recently proven.

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.

Upper Bounds on the Paired Domination Subdivision Number of a Graph

2012 ◽  
Vol 29 (4) ◽  
pp. 843-856
Author(s):  
Yoshimi Egawa ◽  
Michitaka Furuya ◽  
Masanori Takatou
Keyword(s):  
Upper Bounds ◽  
Paired Domination ◽  
Domination Subdivision 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.

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 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).

An Upper Bound for the Total Domination Subdivision Number of a Graph

2009 ◽  
Vol 25 (5) ◽  
pp. 727-733 ◽  
Author(s):  
H. Karami ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami ◽  
A. Khodkar
Keyword(s):  
Upper Bound ◽  
Total Domination ◽  
Total Domination Subdivision Number ◽  
Domination Subdivision Number

Game total domination subdivision number of a graph

2015 ◽  
Vol 07 (03) ◽  
pp. 1550023
Author(s):  
J. Amjadi ◽  
H. Karami ◽  
S. M. Sheikholeslami
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Optimal Strategies ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Domination Subdivision Number ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Domination Subdivision Number

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγt(G) is the minimum cardinality of a total dominating set of G. The game total domination subdivision number of a graph G is defined by the following game. Two players 𝒟 and 𝒜, 𝒟 playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of 𝒟 is to minimize the total domination number γt(G′) of G′ while 𝒜 tries to maximize it. If both 𝒜 and 𝒟 play according to their optimal strategies, γt(G′) is well defined. We call this number the game total domination subdivision number of G and denote it by γgt(G). In this paper we initiate the study of the game total domination subdivision number of a graph and present some (sharp) bounds for this parameter.

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