56 research outputs found
Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems
AbstractIt is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number
The maximum forcing number of polyomino
The forcing number of a perfect matching of a graph is the
cardinality of the smallest subset of that is contained in no other perfect
matchings of . For a planar embedding of a 2-connected bipartite planar
graph which has a perfect matching, the concept of Clar number of hexagonal
system had been extended by Abeledo and Atkinson as follows: a spanning
subgraph of is called a Clar cover of if each of its components is
either an even face or an edge, the maximum number of even faces in Clar covers
of is called Clar number of , and the Clar cover with the maximum number
of even faces is called the maximum Clar cover. It was proved that if is a
hexagonal system with a perfect matching and is a set of hexagons in a
maximum Clar cover of , then has a unique 1-factor. Using this
result, Xu {\it et. at.} proved that the maximum forcing number of the
elementary hexagonal system are equal to their Clar numbers, and then the
maximum forcing number of the elementary hexagonal system can be computed in
polynomial time. In this paper, we show that an elementary polyomino has a
unique perfect matching when removing the set of tetragons from its maximum
Clar cover. Thus the maximum forcing number of elementary polyomino equals to
its Clar number and can be computed in polynomial time. Also, we have extended
our result to the non-elementary polyomino and hexagonal system
Resonance graphs of plane bipartite graphs as daisy cubes
We characterize all plane bipartite graphs whose resonance graphs are daisy
cubes and therefore generalize related results on resonance graphs of benzenoid
graphs, catacondensed even ring systems, as well as 2-connected outerplane
bipartite graphs. Firstly, we prove that if is a plane elementary bipartite
graph other than , then the resonance graph is a daisy cube if and
only if the Fries number of equals the number of finite faces of , which
in turn is equivalent to being homeomorphically peripheral color
alternating. Next, we extend the above characterization from plane elementary
bipartite graphs to all plane bipartite graphs and show that the resonance
graph of a plane bipartite graph is a daisy cube if and only if is
weakly elementary bipartite and every elementary component of other than
is homeomorphically peripheral color alternating. Along the way, we prove
that a Cartesian product graph is a daisy cube if and only if all of its
nontrivial factors are daisy cubes
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max
formula for the maximum cardinality of the union of k sink-stable sets. The
results imply a recent min-max theorem of Abeledo and Atkinson on the Clar
number of bipartite plane graphs and a sharpening of Minty's coloring theorem.
We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on
cyclic stable sets
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty’s coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse ́ and of Sebő on cyclic stable sets
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