8 research outputs found
Maximal Independent Sets In Graphs With At Most r Cycles
We find the maximum number of maximal independent sets in two families of
graphs: all graphs with vertices and at most cycles, and all such
graphs that are also connected. In addition, we characterize the extremal
graphs.Comment: 31 pages, 11 figures, Latex, see related papers at
http://www.math.msu.edu/~sagan, split paper into two part
Counting non-isomorphic maximal independent sets of the n-cycle graph
The number of maximal independent sets of the n-cycle graph C_n is known to
be the nth term of the Perrin sequence. The action of the automorphism group of
C_n on the family of these maximal independent sets partitions this family into
disjoint orbits, which represent the non-isomorphic (i.e., defined up to a
rotation and a reflection) maximal independent sets. We provide exact formulas
for the total number of orbits and the number of orbits having a given number
of isomorphic representatives. We also provide exact formulas for the total
number of unlabeled (i.e., defined up to a rotation) maximal independent sets
and the number of unlabeled maximal independent sets having a given number of
isomorphic representatives. It turns out that these formulas involve both
Perrin and Padovan sequences.Comment: Revised versio
Maximal independent sets and maximal matchings in series-parallel and related graph classes
The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and seriesparallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].Postprint (author's final draft