983 research outputs found
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
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
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
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Subgraph characterization of Red/Blue-split graphs and König-Egerváry graphs
König-Egerváry graphs (KEGs) are the graphs whose maximum size of a matching is equal to the minimum size of a vertex cover. We give an excluded subgraph characterization of KEGs. We show that KEGs are a special case of a more general class of graph: emph{Red/Blue-split} graphs, and give an excluded subgraph characterization of Red/Blue-split graphs. We show several consequences of this result including theorems of Deming-Sterboul, Lovász, and Földes-Hammer. A refined result of Schrijver on the integral solution of certain systems of linear inequalities is also given through the result on the weighted version of Red/Blue-split graphs
A Counterexample Regarding Labelled Well-Quasi-Ordering
Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns
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