15 research outputs found
A Maximum Resonant Set of Polyomino Graphs
A polyomino graph is a connected finite subgraph of the infinite plane
grid such that each finite face is surrounded by a regular square of side
length one and each edge belongs to at least one square. In this paper, we show
that if is a maximum resonant set of , then has a unique perfect
matching. We further prove that the maximum forcing number of a polyomino graph
is equal to its Clar number. Based on this result, we have that the maximum
forcing number of a polyomino graph can be computed in polynomial time. We also
show that if is a maximal alternating set of , then has a unique
perfect matching.Comment: 13 pages, 6 figure
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
Tight upper bound on the maximum anti-forcing numbers of graphs
Let be a simple graph with a perfect matching. Deng and Zhang showed that
the maximum anti-forcing number of is no more than the cyclomatic number.
In this paper, we get a novel upper bound on the maximum anti-forcing number of
and investigate the extremal graphs. If has a perfect matching
whose anti-forcing number attains this upper bound, then we say is an
extremal graph and is a nice perfect matching. We obtain an equivalent
condition for the nice perfect matchings of and establish a one-to-one
correspondence between the nice perfect matchings and the edge-involutions of
, which are the automorphisms of order two such that and
are adjacent for every vertex . We demonstrate that all extremal
graphs can be constructed from by implementing two expansion operations,
and is extremal if and only if one factor in a Cartesian decomposition of
is extremal. As examples, we have that all perfect matchings of the
complete graph and the complete bipartite graph are nice.
Also we show that the hypercube , the folded hypercube ()
and the enhanced hypercube () have exactly ,
and nice perfect matchings respectively.Comment: 15 pages, 7 figure
Maximizing the minimum and maximum forcing numbers of perfect matchings of graphs
Let be a simple graph with vertices and a perfect matching. The
forcing number of a perfect matching of is the smallest
cardinality of a subset of that is contained in no other perfect matching
of . Among all perfect matchings of , the minimum and maximum values
of are called the minimum and maximum forcing numbers of , denoted
by and , respectively. Then . Che and Chen
(2011) proposed an open problem: how to characterize the graphs with
. Later they showed that for bipartite graphs , if and
only if is complete bipartite graph . In this paper, we solve the
problem for general graphs and obtain that if and only if is a
complete multipartite graph or ( with arbitrary additional
edges in the same partite set). For a larger class of graphs with
we show that is -connected and a brick (3-connected and
bicritical graph) except for . In particular, we prove that the
forcing spectrum of each such graph is continued by matching 2-switches and
the minimum forcing numbers of all such graphs form an integer interval
from to
Relations between global forcing number and maximum anti-forcing number of a graph
The global forcing number of a graph G is the minimal cardinality of an edge
subset discriminating all perfect matchings of G, denoted by gf(G). For any
perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M
such that G-S has a unique perfect matching is called the anti-forcing number
of M,denoted by af(G, M). The maximum anti-forcing number of G among all
perfect matchings is denoted by Af(G). It is known that the maximum
anti-forcing number of a hexagonal system equals the famous Fries number.
We are interested in some comparisons between the global forcing number and
the maximum anti-forcing number of a graph. For a bipartite graph G, we show
that gf(G)is larger than or equal to Af(G). Next we mainly extend such result
to non-bipartite graphs, which is the set of all graphs with a perfect matching
which contain no two disjoint odd cycles such that their deletion results in a
subgraph with a perfect matching. For any such graph G, we also have gf(G) is
larger than or equal to Af(G) by revealing further property of non-bipartite
graphs with a unique perfect matching. As a consequence, this relation also
holds for the graphs whose perfect matching polytopes consist of non-negative
1-regular vectors. In particular, for a brick G, de Carvalho, Lucchesi and
Murty [4] showed that G satisfying the above condition if and only if G is
solid, and if and only if its perfect matching polytope consists of
non-negative 1-regular vectors.
Finally, we obtain tight upper and lower bounds on gf(G)-Af(G). For a
connected bipartite graph G with 2n vertices, we have that 0 \leq gf(G)-Af(G)
\leq 1/2 (n-1)(n-2); For non-bipartite case, -1/2 (n^2-n-2) \leq gf(G)-Af(G)
\leq (n-1)(n-2).Comment: 19 pages, 11 figure
Sparse and random sampling techniques for high-resolution, full-field, bss-based structural dynamics identification from video
Video-based techniques for identification of structural dynamics have the advantage that they are very inexpensive to deploy compared to conventional accelerometer or strain gauge techniques. When structural dynamics from video is accomplished using full-field, high-resolution analysis techniques utilizing algorithms on the pixel time series such as principal components analysis and solutions to blind source separation the added benefit of high-resolution, full-field modal identification is achieved. An important property of video of vibrating structures is that it is particularly sparse. Typically video of vibrating structures has a dimensionality consisting of many thousands or even millions of pixels and hundreds to thousands of frames. However the motion of the vibrating structure can be described using only a few mode shapes and their associated time series. As a result, emerging techniques for sparse and random sampling such as compressive sensing should be applicable to performing modal identification on video. This work presents how full-field, high-resolution, structural dynamics identification frameworks can be coupled with compressive sampling. The techniques described in this work are demonstrated to be able to recover mode shapes from experimental video of vibrating structures when 70% to 90% of the frames from a video captured in the conventional manner are removed