# Total domination versus paired domination

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 3, page 435-447
- ISSN: 2083-5892

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topOliver Schaudt. "Total domination versus paired domination." Discussiones Mathematicae Graph Theory 32.3 (2012): 435-447. <http://eudml.org/doc/270836>.

@article{OliverSchaudt2012,

abstract = {A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γₚ. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γₚ.
In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γₚ/γₜ ≤ 2 - 2/r for $K_\{1,r\}$-free graphs. As a consequence, we obtain the sharp bound γₚ/γₜ ≤ 2 - 2/(Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that $\{C₅,T_r\}$-free graphs fulfill the sharp bound γₚ/γₜ ≤ 2 - 2/r, where $T_r$ is obtained from $K_\{1,r\}$ by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γₚ/Γₜ. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γₚ ≤ Γₜ holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γₚ/Γₜ taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γₚ/Γₜ ≤ 2 - 2/r for graphs that do not contain the corona of $K_\{1,r\}$ as subgraph. In particular, we obtain the sharp bound γₚ/Γₜ ≤ 2 - 2/Δ.},

author = {Oliver Schaudt},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {total domination; upper total domination; paired domination; upper paired domination; generalized claw-free graphs},

language = {eng},

number = {3},

pages = {435-447},

title = {Total domination versus paired domination},

url = {http://eudml.org/doc/270836},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Oliver Schaudt

TI - Total domination versus paired domination

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 3

SP - 435

EP - 447

AB - A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γₚ. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γₚ.
In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γₚ/γₜ ≤ 2 - 2/r for $K_{1,r}$-free graphs. As a consequence, we obtain the sharp bound γₚ/γₜ ≤ 2 - 2/(Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that ${C₅,T_r}$-free graphs fulfill the sharp bound γₚ/γₜ ≤ 2 - 2/r, where $T_r$ is obtained from $K_{1,r}$ by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γₚ/Γₜ. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γₚ ≤ Γₜ holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γₚ/Γₜ taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γₚ/Γₜ ≤ 2 - 2/r for graphs that do not contain the corona of $K_{1,r}$ as subgraph. In particular, we obtain the sharp bound γₚ/Γₜ ≤ 2 - 2/Δ.

LA - eng

KW - total domination; upper total domination; paired domination; upper paired domination; generalized claw-free graphs

UR - http://eudml.org/doc/270836

ER -

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