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Characterising and recognising game-perfect graphs
Consider a vertex colouring game played on a simple graph with
permissible colours. Two players, a maker and a breaker, take turns to colour
an uncoloured vertex such that adjacent vertices receive different colours. The
game ends once the graph is fully coloured, in which case the maker wins, or
the graph can no longer be fully coloured, in which case the breaker wins. In
the game , the breaker makes the first move. Our main focus is on the
class of -perfect graphs: graphs such that for every induced subgraph ,
the game played on admits a winning strategy for the maker with only
colours, where denotes the clique number of .
Complementing analogous results for other variations of the game, we
characterise -perfect graphs in two ways, by forbidden induced subgraphs
and by explicit structural descriptions. We also present a clique module
decomposition, which may be of independent interest, that allows us to
efficiently recognise -perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the
International Colloquium on Graph Theory (ICGT) 201
On characterizing game-perfect graphs by forbidden induced subgraphs
A graph is called -perfect if, for any induced subgraph of , the game chromatic number of equals the clique number of . A graph is called -col-perfect if, for any induced subgraph of , the game coloring number of equals the clique number of . In this paper we characterize the classes of -perfect resp. -col-perfect graphs by a set of forbidden induced subgraphs and explicitly. Moreover, we study similar notions for variants of the game chromatic number, namely -perfect and -perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs
Line game-perfect graphs
The -edge colouring game is played with a set of colours on a
graph with initially uncoloured edges by two players, Alice (A) and Bob
(B). The players move alternately. Player has the first move.
. If , then only player may skip any move,
otherwise skipping is not allowed for any player. A move consists in colouring
an uncoloured edge with one of the colours such that adjacent edges have
distinct colours. When no more moves are possible, the game ends. If every edge
is coloured in the end, Alice wins; otherwise, Bob wins.
The -game chromatic index is the smallest
nonnegative integer such that Alice has a winning strategy for the
-edge colouring game played on with colours. The graph is
called line -perfect if, for any edge-induced subgraph of ,
where denotes the clique
number of the line graph of .
For each of the six possibilities , we
characterise line -perfect graphs by forbidden (edge-induced) subgraphs
and by explicit structural descriptions, respectively
Positional games on random graphs
We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability for the
existence of Maker's strategy to claim a member of in the unbiased game
played on the edges of random graph , for various target families
of winning sets. More generally, for each probability above this threshold we
study the smallest bias such that Maker wins the biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game
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