16 research outputs found
On the perfect orderability of unions of two graphs
A graph G is perfectly orderable if it admits an order < on its vertices such that the
sequential coloring algorithm delivers an optimum coloring on each induced subgraph
(H, <) of (G, <). A graph is a threshold graph if it contains no P4 , 2K2 or C4 as
induced subgraph. A theorem of Chvatal, Hoang, Mahadev and de Werra states
that a graph is perfectly orderable if it can be written as the union of two threshold
graphs. In this thesis, we investigate possible generalizations of the above theorem.
We conjecture that if G is the union of two graphs G1 and G2 then G is perfectly
orderable whenever (i) G1 and G2 are both P4 -free and 2K2-free, or (ii) G1 is P4-free,
2K2-free and G2 is P4 -free, C4 -free. We show that the complement of the chordless
cycle with at least five vertices cannot be a counter-example to our conjecture and
we prove, jointly with Hoang, a special case of (i): if G1 and G2 are two edge disjoint
graphs that are P4 -free and 2K2 -free then the union of G1 and G2 is perfectly
orderable
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
Well-quasi-ordering versus clique-width : new results on bigenic classes.
Daligault, Rao and Thomassé conjectured that if a hereditary class of graphs is well-quasi-ordered by the induced subgraph relation then it has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this conjecture is not true for infinitely defined classes. For finitely defined classes the conjecture is still open. It is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4P4. For bigenic classes of graphs i.e. ones defined by two forbidden induced subgraphs there are several open cases in both classifications. We reduce the number of open cases for well-quasi-orderability of such classes from 12 to 9. Our results agree with the conjecture and imply that there are only two remaining cases to verify for bigenic classes
Well-quasi-ordering versus clique-width: new results on bigenic classes
Daligault, Rao and Thomassé conjectured that if a hereditary class of graphs is well-quasi-ordered by the induced subgraph relation then it has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this conjecture is not true for infinitely defined classes. For finitely defined classes the conjecture is still open. It is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4P4. For bigenic classes of graphs i.e. ones defined by two forbidden induced subgraphs there are several open cases in both classifications. We reduce the number of open cases for well-quasi-orderability of such classes from 12 to 9. Our results agree with the conjecture and imply that there are only two remaining cases to verify for bigenic classes
A parametric approach to hereditary classes
The “minimal class approach" consists of studying downwards-closed properties of hereditary graph classes (such as boundedness of a certain parameter within the class) by identifying the minimal obstructions to those properties. In this thesis, we look at various hereditary classes through this lens. In practice, this often amounts to analysing the structure of those classes by characterising boundedness of certain graph parameters within them. However, there is more to it than this: while adopting the minimal class viewpoint, we encounter a variety of interesting notions and problems { some more loosely related to the approach than others. The thesis compiles the author's work in the ensuing research directions