Independent Edges Research Articles

Group Connectivity and Matchings

Let $$\alpha ^\prime (G)$$??(G) be the maximum number of independent edges in a graph $$G$$G and $$A$$A an abelian group with $$|A|\ge 4$$|A|?4. If a graph $$G^*$$G? is obtained by repeatedly contracting nontrivial $$A$$A-connected subgraphs of $$G$$G until no such a subgraph left, then we say $$G$$G can be $$A$$A-reduced to $$G^*$$G?. In this paper, we show that if $$G$$G is a 3-edge-connected simple graph and $$\alpha ^\prime (G)\le 5$$??(G)≤5, then either $$G$$G is $$A$$A-connected or $$G$$G can be $$A$$A-reduced to the Petersen graph.

Color schlieren imaging with a two-path, double knife edge system

The traditional arrangement for visualizing optical phenomena with the schlieren technique is modified to include a Mach-Zehnder geometry. This allows for the implementation of two independent knife edges along two different beam paths, resulting in an enhanced combined image that is uniquely adjustable. Post-processed combined images are also generated by spatially separating the paths from each arm and then colorizing and combining the images into a single composite. In this way, bidirectional, color schlieren images have been produced using both white-light and monochromatic sources.

Combinatorial Conditions for the Unique Completability of Low-Rank Matrices

We consider the problems of completing a low-rank positive semidefinite square matrix $M$ or a low-rank rectangular matrix $N$ from a given subset of their entries. Following the approach initiated by Singer and Cucuringu [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1621--1641] we study the local and global uniqueness of such completions by analyzing the structure of the graphs determined by the positions of the known entries of $M$ or $N$. We present combinatorial characterizations of local and global (unique) completability for special families of graphs. We characterize local and global completability in all dimensions for cluster graphs, i.e. graphs which can be obtained from disjoint complete graphs by adding a set of independent edges. These results correspond to theorems for body-bar frameworks in rigidity theory. We also provide a characterization of two-dimensional local completability of planar bipartite graphs, which leads to a characterization of two-dimensional local completability in the rectangular matrix model when the underlying bipartite graph is planar. These results are based on new observations that certain graph operations preserve local or global completability, as well as on a further connection between rigidity and completability. We also prove that a rank condition on the completability stress matrix of a graph is a sufficient condition for global completability. This verifies a conjecture of Singer and Cucuringu given in the paper cited above.

Invariant Kekulé structures in fullerene graphs

Abstract Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekule structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekule structures could have special physical and chemical properties, as suggested in [Austin, S.J, and J. Baker, P. W. Fowler, D. E. Manolopoulos, Bond-stretch Isomerism and the Fullerenes, J. Chem. Soc. Perkin Trans. 2 (1994), 2319–2323] and [Rogers, K.M., and P. W. Fowler, Leapfrog fullerenes, Huckel bond order and Kekule structures, J. Chem. Soc. Perkin Trans. 2 (2001), 18–22]. All fullerenes with at least ten symmetries were studied in [Graver, J.E. The Structure of Fullerene Signature, DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64, AMS (2005), 137–166.] and a complete catalog was given in [Graver, J. E. Catalog of All Fullerene with Ten or More Symmetries DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), 167–188]. Starting from this catalog in [Bogaerts, M., and G. Mazzuoccolo, G.Rinaldi, Totally symmetric Kekule structures in fullerene graphs with ten or more symmetries, MATCH Communications in Mathematical and in Computer Chemistry 69 (2013), 677–705] we established exactly which of them have at least one totally symmetric Kekule structure.

The number of Kekulé structures of polyominos on the torus

Let G be a (molecule) graph. A perfect matching, or Kekule structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekule structures of a (molecule) graph is interest in chemistry, physics and mathematics. In this paper, we focus on some polyominos on the torus and obtain the explicit expressions on the number of the Kekule structures of them.

A Contribution to the Second Neighborhood Problem

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.

Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij })

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij})

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Bernoulli Graph Bounds for Generalrandom Graphs

General random graphs (i.e., stochastic models for networks incorporating heterogeneity and/or dependence among edges) are increasingly in wide use in the study of social and other networks, but few techniques other than simulation have been available for studying their behavior. On the other hand, random graphs with independent edges (i.e., the Bernoulli graphs) are well-studied, and a large literature exists regarding their properties. In this paper, we demonstrate a method for leveraging this knowledge by constructing families of Bernoulli graphs that bound the behavior of an arbitrary random graph in a well-defined sense. By studying the behavior of these Bernoulli graph bounds, we can thus constrain the properties of a given random graph. We illustrate the utility of this approach via application to several problems from the social network literature, including identifying degeneracy in Markov graph models, studying the potential impact of tie formation mechanisms on epidemic potential in sexual contact networks, and robustness testing of inhomogeneous Bernoulli models based on geographical covariates. Practical heuristics for assessing bound tightness and guidance for use in theoretical and methodological applications are also discussed.

Linear-time algorithm for the matched-domination problem in cographs

Let G=(V, E) be a graph without isolated vertices. A matching in G is a set of independent edges in G. A perfect matching M in G is a matching such that every vertex of G is incident to an edge of M. A set S⊆ V is a paired-dominating set of G if every vertex not in S is adjacent to a vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination problem is to find a paired-dominating set of G with minimum cardinality. This paper introduces a generalization of the paired-domination problem, namely, the matched-domination problem, in which some constrained vertices are in paired-dominating sets as far as they can. Further, possible applications are also presented. We then present a linear-time constructive algorithm to solve the matched-domination problem in cographs.† †A preliminary version of this paper has appeared in: Proceedings of the 4th IASTED International Conference on Computational Intelligence (CI’2009), Honolulu, Hawaii, USA, 2009, pp. 120–126.

Supereulerian graphs and matchings

A graph G is called supereulerian if G has a spanning Eulerian subgraph. Let α ′ ( G ) be the maximum number of independent edges in the graph G . In this paper, we show that if G is a 2-edge-connected simple graph and α ′ ( G ) ≤ 2 , then G is supereulerian if and only if G is not K 2 , t for some odd number t .

Formula omitted]-alternating paths and the construction of defect [formula omitted]-extendable bipartite graphs with different connectivities

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n ≤ ( | V ( G ) | − 2 ) / 2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n - extendable. In this paper, we first characterize defect n -extendable bipartite graph G with n = 1 or κ ( G ) ≥ 2 respectively using M -alternating paths. Furthermore, we present a construction characterization of defect n -extendable bipartite graph G with n ≥ 2 and κ ( G ) = 1 . It is also shown that these characterizations can be transformed to polynomial time algorithms to determine if a given bipartite graph is defect n -extendable.

Equivalence of a random intersection graph andG(n,p)

AbstractWe solve the conjecture of Fill, Scheinerman and Singer‐Cohen (Random Struct Algorithms 16 (2000), 156–176) and show equivalence of sharp threshold functions of a random intersection graph${\cal g}$(n,m,p) withm≥n3and a graphG(n,p̂) with independent edges. Moreover we prove sharper equivalence results under some additional assumptions. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Simulative Evaluation of SNR in Asynchronous Sigma-Delta ADC

This paper shows the difference between the SNR level for classical synchronous analog-to-digital converters (ADC) and asynchronous Sigma-Delta analog-to-digital converters (ASD-ADC). Simulative evaluation shows that conversion quality for ASD-ADC is 2 bit worse than for classical synchronous ADCs. The reason of such difference is at first the nonlinearity of ASD-ADC sampling block (ASDM) and at second quantization noise bigger than in classical ADCs. The quantization noise is caused by the need of two independent pulse edges time position measurements. The SNR is evaluated for three stochastic models of input signals: Gauss, uniform, sinusoidal.

Minimum degree of minimal defect [formula omitted]-extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n -extendable. If for any edge e in a defect n -extendable graph G , G − e is not defect n -extendable, then G is minimal defect n -extendable. The minimum degree and the connectivity of a graph G are denoted by δ ( G ) and κ ( G ) respectively. In this paper, we study the minimum degree of minimal defect n -extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has δ ( G ) = 1 . Consider a minimal defect n -extendable bipartite graph G with n ≥ 2 , we show that if κ ( G ) = 1 , then δ ( G ) ≤ n + 1 and if κ ( G ) ≥ 2 , then 2 ≤ δ ( G ) = κ ( G ) ≤ n + 1 . In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds.

Construction characterizations for defect n-extendable bipartite graphs

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n (|V(G)|–2)/2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. This paper presents two construction characterizations of defect n-extendable bipartite graphs and a necessary condition for minimal defect n-extendable bipartite graphs.

2-extendability of toroidal polyhexes and Klein-bottle polyhexes

A toroidal polyhex (resp. Klein-bottle polyhex) described by a string ( p , q , t ) arises from a p × q -parallelogram of a hexagonal lattice by a usual torus (resp. Klein bottle) boundary identification with a torsion t . A connected graph G admitting a perfect matching is k -extendable if | V ( G ) | ≥ 2 k + 2 and any k independent edges can be extended to a perfect matching of G . In this paper, we characterize 2-extendable toroidal polyhexes and 2-extendable Klein-bottle polyhexes.

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V 1,V 2;E) be a bipartite graph with |V 1|=|V 2|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥⌈(4n+2k−1)/3⌉ for each pair of nonadjacent vertices x and y of G with x∈V 1 and y∈V 2, then, for any k independent edges e 1,…,e k of G, G contains k vertex-disjoint quadrilaterals C 1,…,C k such that e i ∈E(C i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V 1|=|V 2|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C 1,…,C k containing specified edges of G.

On 2-factors with cycles containing specified edges in a bipartite graph

Let k ≥ 1 be an integer and G = ( V 1 , V 2 ; E ) a bipartite graph with | V 1 | = | V 2 | = n such that n ≥ 2 k + 2 . In this paper it has been proved that if for each pair of nonadjacent vertices x ∈ V 1 and y ∈ V 2 , d ( x ) + d ( y ) ≥ ⌈ 4 n + 2 k − 1 3 ⌉ , then for any k independent edges e 1 , … , e k of G , G has a 2-factor with k + 1 cycles C 1 , … , C k + 1 such that e i ∈ E ( C i ) and | V ( C i ) | = 4 for each i ∈ { 1 , … , k } . We shall also show that the conditions in this paper are sharp.

Characterizing defect n-extendable graphs and [formula omitted]-critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n ⩽ ( | V ( G ) | - 2 ) / 2 , then G is said to be defect n-extendable. If deleting any k vertices in G where k ⩽ | V ( G ) | - 2 , the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n-extendable graphs and ( 2 k + 1 ) -critical graphs using M-alternating paths are presented.