The game of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi mathvariant="script">F</mml:mi></mml:math>-saturator

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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The Game Saturation Number of a Graph

Carraher

Kinnersley

Reiniger

et al. 2016

Journal of Graph Theory

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Game saturation of intersecting families

Patkós

Vizer

2014

Open Mathematics

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Abstract:We consider the following combinatorial game: two players, Fast and Slow, claim -element subsets of [ ] = {1 2 } alternately, one at each turn, so that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed -subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation numbers, gsat F (I ) and gsat S (I ), are the score of the game when both players play according to an optimal strategy in the cases when the game starts with Fast's or Slow's move, respectively. We prove that MSC:05D05, 91A24

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Graph saturation in multipartite graphs

Ferrara

Jacobson

Pfender

et al. 2016

Journal of Combinatorics

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Let G be a fixed graph and let F be a family of graphs. A subgraph J of G is F-saturated if no member of F is a subgraph of J, but for any edge e in E(G) − E(J), some element of F is a subgraph of J + e. We let ex(F, G) and sat(F, G) denote the maximum and minimum size of an F-saturated subgraph of G, respectively. If no element of F is a subgraph of G, then sat(In this paper, for k ≥ 3 and n ≥ 100 we determine sat(K 3 , K n k ), where K n k is the complete balanced k-partite graph with partite sets of size n. We also give several families of constructions of K t -saturated subgraphs of K n k for t ≥ 4. Our results and constructions provide an informative contrast to recent results on the edge-density

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The game of F-saturator

Cited by 3 publications

References 12 publications

The Game Saturation Number of a Graph

The Game Saturation Number of a Graph

Game saturation of intersecting families

Graph saturation in multipartite graphs

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