5,949 research outputs found
Complements of nearly perfect graphs
A class of graphs closed under taking induced subgraphs is -bounded if
there exists a function such that for all graphs in the class, . We consider the following question initially studied in [A.
Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em
Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a
-bounded class , is the class -bounded (where
is the class of graphs formed by the complements of graphs from
)? We show that if is -bounded by the constant function
, then is -bounded by
and this is best possible. We show that for
every constant , if is -bounded by a function such that
for , then is -bounded. For every ,
we construct a class of graphs -bounded by whose
complement is not -bounded
On self-complementation
We prove that, with very few exceptions, every graph of order n, n - 0, 1(mod 4) and size at most n - 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Internal Partitions of Regular Graphs
An internal partition of an -vertex graph is a partition of
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every -regular graph with
vertices has an internal partition. Here we prove this for . The case
is of particular interest and leads to interesting new open problems on
cubic graphs. We also provide new lower bounds on and find new families
of graphs with no internal partitions. Weighted versions of these problems are
considered as well
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