Game matching number of graphs

Cranston, Daniel W.; Kinnersley, William B.; Suil, O; West, Douglas B.

doi:10.1016/j.dam.2013.03.010

Cited by 14 publications

(13 citation statements)

References 15 publications

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“…Min wins Y (8). If after four edges the selected graph contains a triangle, then the remaining game is X (5) in which the first move has been played, which is won by Min. To avoid this, Max must make 2K 2 , and after Min makes P 3 + P 2 Max must make P 3 + 2P 2 , extend one of the paths, or join the paths to form P 5 .…”

Section: Theorem 22mentioning

confidence: 99%

“…To avoid this, Max must make 2K 2 , and after Min makes P 3 + P 2 Max must make P 3 + 2P 2 , extend one of the paths, or join the paths to form P 5 . If P 3 + 2P 2 , then again Min closes the triangle, and the remainder is a copy of X (5) in which Min has played a winning 2K 2 . In the other cases, Min forms P 6 ; the final graph is C 6 + P 2 or C 7 + P 1 , with Min winning.…”

Section: Theorem 22mentioning

confidence: 99%

“…The P 3 ‐saturation game was studied by Cranston, Kinnersley, O, and West ; here

P_{k}

denotes the k ‐vertex path. The subgraphs of H that are P 3 ‐saturated relative to H are precisely the maximal matchings in H .…”

Section: Introductionmentioning

confidence: 99%

“…We have mentioned bounds on

α_{g}^{'}

but not

{\hat{α}}_{g}^{'}

because the two parameters never differ by more than 1 (see ). This does not hold for

scriptF

‐saturation in general.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Game Saturation Number of a Graph

Carraher

Kinnersley

Reiniger

et al. 2016

Journal of Graph Theory

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Given a family scriptF and a host graph H, a graph G⊆H is scriptF‐saturated relative to H if no subgraph of G lies in scriptF but adding any edge from E(H)−E(G) to G creates such a subgraph. In the scriptF‐saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in scriptF, until G becomes scriptF‐saturated relative to H. They aim to maximize or minimize the length of the game, respectively; sat gfalse(scriptF;Hfalse) denotes the length under optimal play (when Max starts). Let scriptO denote the family of odd cycles and Tn the family of n‐vertex trees, and write F for scriptF when F={F}. Our results include sat gfalse(scriptO;Knfalse)=⌊n2⌋false⌈n2false⌉, sat gfalse(Tn;Knfalse)=false(0ptn−22false)+1 for n≥6, sat gfalse(K1,3;Knfalse)=2⌊n2⌋ for n≥8, and sat gfalse(P4;Knfalse)∈{⌊4n5⌋,⌈4n5⌉} for n≥5. We also determine sat gfalse(P4;Km,nfalse); with m≥n, it is n when n is even, m when n is odd and m is even, and m+⌊n/2⌋ when mn is odd. Finally, we prove the lower bound sat gfalse(C4;Kn,nfalse)≥121n13/12−Ofalse(n35/36false). The results are very similar when Min plays first, except for the P4‐saturation game on Km,n.

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Section: Theorem 22mentioning

confidence: 99%

Section: Theorem 22mentioning

confidence: 99%

“…The P 3 ‐saturation game was studied by Cranston, Kinnersley, O, and West ; here

P_{k}

denotes the k ‐vertex path. The subgraphs of H that are P 3 ‐saturated relative to H are precisely the maximal matchings in H .…”

Section: Introductionmentioning

confidence: 99%

“…We have mentioned bounds on

α_{g}^{'}

but not

{\hat{α}}_{g}^{'}

because the two parameters never differ by more than 1 (see ). This does not hold for

scriptF

‐saturation in general.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Game Saturation Number of a Graph

Carraher

Kinnersley

Reiniger

et al. 2016

Journal of Graph Theory

Self Cite

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“…[15]) and independently by Bodlaender [2] for general graphs; it has seen extensive study, see the survey [28]. Recently, work has been done on competitive optimization variants of list-colouring [5] and its more studied related version called paintability as introduced in [26] (for further references see Section 8 of [28]), matching [11], domination [4], total domination [19], disjoint domination [8], Ramsey theory [9,16,17], and more [3].…”

Section: Introductionmentioning

confidence: 99%

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Bujtás¹,

Henning²,

Tuza³

2016

SIAM J. Discrete Math.

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The 3 4 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γ tg (G) ≤ 3 4 n, where γ tg (G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τ g (H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H C 4 , then τ g (H) ≤ 4 11 (n H + m H ), where n H and m H denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γ tg (G) < 8 11 n. As a consequence of this result, the 3 4 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.

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