Gallai-Edmonds Structure Theorem Research Articles

Nordhaus–Gaddum type inequality for the integer k-matching number of a graph

An integer k-matching of a graph G is a function h:E(G)→{0,1,…,k} such that ∑e∈Γ(v)h(e)≤k for any v∈V(G), where Γ(v) is the set of edges incident to v. The integer k-matching number of G, denoted by mk(G), is the maximum number of ∑e∈E(G)h(e) over all integer k-matching h of G. In this paper, we establish the following lower bounds on the sum of the integer k-matching number of a graph G and its complement by using Gallai–Edmonds Structure Theorem: (1) mk(G)+mk(G¯)≥⌊nk2⌋ for n≥2;(2) if G and G¯ are non-empty, then for n≥25, mk(G)+mk(G¯)≥⌊nk+k2⌋;(3) if G and G¯ have no isolated vertices, then for n≥25, mk(G)+mk(G¯)≥⌊nk2⌋+2k . Furthermore, all extremal graphs attaining the lower bounds are also characterized.

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.

Critical independent sets of König–Egerváry graphs

Let G be a simple graph with vertex set V ( G ) and edge set E ( G ) . For x ∈ V ( G ) , let N G ( x ) = { y ∈ V ( G ) | x y ∈ E ( G ) } and N G ( X ) = ⋃ x ∈ X N G ( x ) . For X ⊆ V ( G ) , the number d ( X ) ≔ | X | − | N G ( X ) | is called difference of X and critical difference of G denoted by d c ( G ) is max { d ( X ) ∣ X ⊆ V ( G ) } . A set S ⊆ V ( G ) is a critical independent set if S is an independent set and d ( S ) = d c ( G ) . Zhang (1990) proved that for every graph, there exists an independent set attaining the critical difference. Let Λ 0 ( G ) denote the family of all nonempty critical independent sets of G . If α ( G ) + μ ( G ) = | V ( G ) | , then G is called a König–Egerváry graph, where α ( G ) and μ ( G ) denote independence number and matching number of G , respectively. We define k e r 0 ( G ) to be the intersection of all sets in Λ 0 ( G ) and c o r e ( G ) to be the intersection of all maximum independent sets. In this paper, we characterize König–Egerváry graphs satisfying k e r 0 ( G ) = c o r e ( G ) in terms of Gallai–Edmonds Structure Theorem. As a corollary of this result, we solve an open problem posed by Jarden et al. (2018).

Regular graphs with equal matching number and independence number

Let r≥3 be an integer and G be a graph. Let δ(G),Δ(G), α(G), and μ(G) denote the minimum degree, maximum degree, independence number, and matching number of G, respectively. Recently, Caro, Davila and Pepper proved δ(G)α(G)≤Δ(G)μ(G). Mohr and Rautenbach characterized the extremal graphs within the classes of all non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs within the classes of all r-regular graphs in terms of the Gallai–Edmonds Structure Theorem, which extends Mohr and Rautenbach’s result.

Atoms of the matching measure

We prove that the matching measure of an infinite vertex-transitive connected graph has no atoms. Generalizing the results of Salez, we show that for an ergodic non-amenable unimodular random rooted graph with uniformly bounded degrees, the matching measure has only finitely many atoms. Ku and Chen proved the analogue of the Gallai-Edmonds structure theorem for non-zero roots of the matching polynomial for finite graphs. We extend their results for infinite graphs. We also show that the corresponding Gallai-Edmonds decomposition is compatible with the zero temperature monomer-dimer model.

On the permanental nullity and matching number of graphs

For a graph G with n vertices, let and A(G) denote the matching number and adjacency matrix of G, respectively. The permanental polynomial of G is defined as . The permanental nullity of G, denoted by , is the multiplicity of the zero root of . In this paper, we use the Gallai–Edmonds structure theorem to derive a concise formula which reveals the relationship between the permanental nullity and the matching number of a graph. Furthermore, we prove a necessary and sufficient condition for a graph G to have . As applications, we show that every unicyclic graph G on n vertices satisfies , that the permanental nullity of the line graph of a graph is either zero or one and that the permanental nullity of a factor critical graph is always zero.

Matching structure and bargaining outcomes in buyer–seller networks

We examine the relationship between the matching structure of a bipartite (buyer–seller) network and the (expected) shares of the unit surplus that each connected pair in this network can create. We show that in different bargaining environments, these shares are closely related to the Gallai–Edmonds Structure Theorem. This theorem characterizes the structure of maximum matchings in an undirected graph. We show that the relationship between the (expected) shares and the Structure Theorem is not an artefact of a particular bargaining mechanism or trade centralization. However, this relationship does not necessarily generalize to non-bipartite networks or to networks with heterogeneous link values.

AN ALTERNATIVE PROOF OF LOVASZ'S CATHEDRAL THEOREM

A graph G with a perfect matching is called saturated if G+e has more perfect matchings than G for any edge e that is not in G. Lovasz gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which revealed canonical structures of general graphs with perfect matchings. Here, the cathedral theorem is derived in quite a natural way, providing more refined or generalized properties. Moreover, the new proof shows that it can be proved without using the Gallai-Edmonds structure theorem.

Gallai–Edmonds structure theorem for weighted matching polynomial

In this paper, we prove the Gallai–Edmonds structure theorem for weighted matching polynomials. Our result implies the Parter–Wiener theorem and its recent generalization about the existence of principal submatrices of a Hermitian matrix whose graph is a tree.

Gallai-Edmonds decomposition as a pruning technique

We introduce a generic propagation mechanism for constraint programming. The method is based on the results of matching theory which is a mature and well-studied subject of graph theory. A first benefit of our new pruning technique comes from the fact that it can be applied on several global constraints whose solution is representable by a matching in a particular graph. In this work we describe a filtering scheme for such a family based on the Gallai-Edmonds Structure Theorem. In a number of important cases our method achieves hyper-arc consistency in polynomial time.

A short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem

We present a short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem from Hall’s Theorem.

Maximum Multiplicity of Matching Polynomial Roots and Minimum Path Cover in General Graphs

Let $G$ be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial $\mu(G,x)$ is at most the minimum number of vertex disjoint paths needed to cover the vertex set of $G$. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, we extend the notion of a $(\theta,G)$-extremal path cover (where $\theta$ is a root of $\mu(G,x)$) which was first introduced for trees to general graphs. Our proof makes use of the analogue of the Gallai-Edmonds Structure Theorem for general root. By way of contrast, we also show that the difference between the minimum size of a path cover and the maximum multiplicity of matching polynomial roots can be arbitrarily large.

An analogue of the Gallai–Edmonds Structure Theorem for non-zero roots of the matching polynomial

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing “deficiency” with multiplicity of θ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai–Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots.

A modified greedy algorithm for dispersively weighted 3-set cover

The set cover problem is that of computing a minimum weight subfamily F ′ , given a family F of weighted subsets of a base set U, such that every element of U is covered by some subset in F ′ . The k-set cover problem is a variant in which every subset is of size at most k. It has been long known that the problem can be approximated within a factor of H ( k ) = ∑ i = 1 k ( 1 / i ) by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted 3-set cover problem, as a first step towards better approximation of general k-set cover problem, where any two distinct subset costs differ by a multiplicative factor of at least 2. It will be shown, via LP duality, that an improved approximation bound of H ( 3 ) - 1 / 6 can be attained, when the greedy heuristic is suitably modified for this case. A key to our algorithm design and analysis is the Gallai–Edmonds structure theorem for maximum matchings.

A generalisation of matching and colouring

We define an r- bounded cover of a graph G to be a subgraph T⊆ G such that T contains all vertices of G and each component of T is a complete subgraph of G of order at most r. A 2-bounded cover of a graph G corresponds to a matching of G, and an ω( G)-bounded cover of G corresponds to a colouring of the vertices of the complement G ̄ . We generalise a number of results on matching and colouring of graphs to r-bounded covers, including the Gallai–Edmonds Structure Theorem, Tutte's 1-Factor Theorem, and Gallai's theorem on the minimal order of colour-critical graphs with connected complements.

A Gallai–Edmonds‐type structure theorem for path‐matchings

AbstractAs a generalization of matchings, Cunningham and Geelen introduced the notion of path‐matchings. We give a structure theorem for path‐matchings which generalizes the fundamental Gallai–Edmonds structure theorem for matchings. Our proof is purely combinatorial. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 93–102, 2004

The fractional matching numbers of graphs

AbstractA fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that, for each vertex v, ∑ f(e) ≤ 1, where the sum is taken over all edges incident to v. The fractional matching number of G is the supremum of ∑e∈E(G) f(e) over all fractional matchings f. In this paper, we provide a new formula for calculating the fractional matching numbers of graphs using the Gallai–Edmonds Structure Theorem. Thus, we characterize graphs for which the fractional matching number equals the matching number and graphs for which the fractional matching number is the maximum possible (one‐half the number of vertices). © 2002 Wiley Periodicals, Inc.

Optimal packing of induced stars in a graph

We consider simple undirected graphs. An edge subset A of G is called an induced n-star packing of G if every component of the subgraph G[ A] induced by A is a star with at most n edges and is an induced subgraph of G. We consider the problem of finding an induced n-star packing of G that covers the maximum number of vertices. This problem is a natural generalization of the classical matching problem. We show that many classical results on matchings (such as the Tutte 1-Factor Theorem, the Berge Duality Theorem, the Gallai-Edmonds Structure Theorem, the Matching Matroid Theorem) can be extended to induced n-star packings in a graph.

A structure theorem for maximum internal matchings in graphs

A counterpart of the Gallai-Edmonds Structure Theorem is proved for matchings that are not required to cover the external vertices of graphs. The appropriate counterpart of Tutte's Theorem is derived from this result for the existence of perfect internal matchings in graphs.