3 research outputs found
Black-and-white threshold graphs
Let k be a natural number. We introduce k-threshold graphs. We show that
there exists an O(n^3) algorithm for the recognition of k-threshold graphs for
each natural number k. k-Threshold graphs are characterized by a finite
collection of forbidden induced subgraphs. For the case k=2 we characterize the
partitioned 2-threshold graphs by forbidden induced subgraphs. We introduce
restricted -, and special 2-threshold graphs. We characterize both classes by
forbidden induced subgraphs. The restricted 2-threshold graphs coincide with
the switching class of threshold graphs. This provides a decomposition theorem
for the switching class of threshold graphs
On the Grundy number of Cameron graphs
The Grundy number of a graph is the maximal number of colors attained by a
first-fit coloring of the graph. The class of Cameron graphs is the Seidel
switching class of cographs. In this paper we show that the Grundy number is
computable in polynomial time for Cameron graphs
Spanning tree enumeration via triangular rank-one perturbations of graph Laplacians
We present new short proofs of known spanning tree enumeration formulae for
threshold and Ferrers graphs by showing that the Laplacian matrices of such
graphs admit triangular rank-one perturbations. We then characterize the set of
graphs whose Laplacian matrices admit triangular rank-one perturbations as the
class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil.
Our work introduces (1) a new characterization of special 2-threshold graphs
that generalizes the characterization of threshold graphs in terms of isolated
and dominating vertices, and (2) a spanning tree enumeration formula for
special 2-threshold graphs that reduces to the aforementioned formulae for
threshold and Ferrers graphs. We consider both unweighted and weighted spanning
tree enumeration.Comment: 21 pages, 14 figure