281 research outputs found
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
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
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
Enumerating Maximal Induced Subgraphs
Given a graph G, the maximal induced subgraphs problem asks to enumerate all maximal induced subgraphs of G that belong to a certain hereditary graph class. While its optimization version, known as the minimum vertex deletion problem in literature, has been intensively studied, enumeration algorithms were only known for a few simple graph classes, e.g., independent sets, cliques, and forests, until very recently [Conte and Uno, STOC 2019]. There is also a connected variation of this problem, where one is concerned with only those induced subgraphs that are connected. We introduce two new approaches, which enable us to develop algorithms that solve both variations for a number of important graph classes. A general technique that has been proven very powerful in enumeration algorithms is to build a solution map, i.e., a multiple digraph on all the solutions of the problem, and the key of this approach is to make the solution map strongly connected, so that a simple traversal of the solution map solves the problem. First, we introduce retaliation-free paths to certify strong connectedness of the solution map we build. Second, generalizing the idea of Cohen, Kimelfeld, and Sagiv [JCSS 2008], we introduce an apparently very restricted version of the maximal (connected) induced subgraphs problem, and show that it is equivalent to the original problem in terms of solvability in incremental polynomial time. Moreover, we give reductions between the two variations, so that it suffices to solve one of the variations for each class we study. Our work also leads to direct and simpler proofs of several important known results
More on discrete convexity
In several recent papers some concepts of convex analysis were extended to
discrete sets. This paper is one more step in this direction. It is well known
that a local minimum of a convex function is always its global minimum. We
study some discrete objects that share this property and provide several
examples of convex families related to graphs and to two-person games in normal
form
Linear rank-width and linear clique-width of trees
We show that for every forest T the linear rank-width of T is equal to the path-width of T, and the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k
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