Ear Decomposition Research Articles

Vector Spaces and the Petersen Graph

It is shown that a matching covered graph has an ear decomposition with no more than one double ear if and only if there is no set $S$ of edges such that $|S \cap A|$ is even for every alternating circuit $A$ but $|S \cap C|$ is odd for some even circuit $C$. Two proofs are presented. The first uses vector spaces and the second is constructive. Some applications are also given.

Finding coherent cyclic orders in strong digraphs

A cyclic order in the vertex set of a digraph is said to be coherent if any arc is contained in a directed cycle whose winding number is one. This notion plays a key role in the proof by Bessy and Thomasse (2004) of a conjecture of Gallai (1964) on covering the vertex set by directed cycles. This paper presents an efficient algorithm for finding a coherent cyclic order in a strongly connected digraph, based on a theorem of Knuth (1974). With the aid of ear decomposition, the algorithm runs in O(nm) time, where n is the number of vertices and m is the number of arcs. This is as fast as testing if a given cyclic order is coherent.

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ n arrangements.

G-Elements, finite buildings and higher Cohen–Macaulay connectivity

Chari proved that if Δ is a ( d − 1 ) -dimensional simplicial complex with a convex ear decomposition, then h 0 ⩽ ⋯ ⩽ h ⌊ d / 2 ⌋ [M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997) 3925–3943]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M-vector [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548]. This is proved to be true by showing that the set of pairs ( ω , Θ ) , where Θ is a l.s.o.p. for k [ Δ ] , the face ring of Δ, and ω is a g-element for k [ Δ ] / Θ , is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag h-vector of such spaces similar in spirit to those examined in [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548] for order complexes of geometric lattices. This also leads to connections between higher Cohen–Macaulay connectivity and conditions which insure that h 0 < ⋯ < h i for a predetermined i.

An O ( VE ) algorithm for ear decompositions of matching-covered graphs

Our main result is an O(nm) -time (deterministic) algorithm for constructing an ear decomposition of a matching-covered graph, where n and m denote the number of nodes and edges. The improvement in the running time comes from new structural results that give a sharpened version of Lovász and Plummer's Two-Ear Theorem. Our algorithm is based on O(nm) -time algorithms for two other fundamental problems in matching theory, namely, finding all the allowed edges of a graph, and finding the canonical partition of an elementary graph. To the best of our knowledge, no faster deterministic algorithms are known for these two fundamental problems.

Connected τ -critical hypergraphs of minimal size

A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) &lt; τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -critical if and only if it is $χ$ -critical 3-chromatic. Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements.

Chain Decompositions of 4-Connected Graphs

In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an $O(|V(G)|^2|E(G)|)$ algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to $O(|V(G)|^3)$.This decomposition will be used to find four independent spanning trees in a 4-connected graph.

On the number of dissimilar pfaffian orientations of graphs

A subgraph H of a graph G is conformal if G - V(H) has a perfect matching. An orientation D of G is Pfaffian if, for every conformal even circuit C , the number of edges of C whose directions in D agree with any prescribed sense of orientation of C is odd. A graph is Pfaffian if it has a Pfaffian orientation. Not every graph is Pfaffian. However, if G has a Pfaffian orientation D , then the determinant of the adjacency matrix of D is the square of the number of perfect matchings of G . (See the book by Lovasz and Plummer [Matching Theory . Annals of Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.] A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. The study of Pfaffian orientations of graphs can be naturally reduced to matching covered graphs. The properties of matching covered graphs are thus helpful in understanding Pfaffian orientations of graphs. For example, say that two orientations of a graph are similar if one can be obtained from the other by reversing the orientations of all the edges in a cut of the graph. Using one of the theorems we proved in [M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, Optimal ear decompositions of matching covered graphs. J. Combinat. Theory B 85 (2002) 59–93] concerning optimal ear decompositions, we show that if a matching covered graph is Pfaffian then the number of dissimilar Pfaffian orientations of G is 2b(G) , where b(G) is the number of “bricks” of G . In particular, any two Pfaffian orientations of a bipartite graph are similar. We deduce that the problem of determining whether or not a graph is Pfaffian is as difficult as the problem of determining whether or not a given orientation is Pfaffian, a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs. Discrete Appl. Math. 25 (1989) 179–180]. We establish a simple property of minimal graphs without a Pfaffian orientation and use it to give an alternative proof of the characterization of Pfaffian bipartite graphs due to Little [ A characterization of convertible (0,1) -matrices. J. Combinat. Theory B 18 (1975) 187–208] .

Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if Δ(L) is the order complex of a rank (r + 1) geometric lattice L, then for all i ≤ r/2 the h-vector of Δ(L) satisfies hi-1 ≤ hi and hi ≤ hr-i. We also obtain several inequalities for the flag h-vector of Δ(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling---Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of hi-1 ≤ hi when i ≤ (2/7)(r + (5/2)).

On finding an ear decomposition of an undirected graph distributively

On finding an ear decomposition of an undirected graph distributively

An Ear Decomposition Approach to Approximating the Smallest 3-Edge Connected Spanning Subgraph of a Multigraph

This paper gives a 3/2 approximation algorithm for the smallest 3-edge connected spanning subgraph of an undirected multigraph. The previous best algorithm of Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] has approximation ratio $5/3$. The algorithm of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528--560] achieves ratio 3/2 for simple graphs. Our approach, based on the close relationship between an ear decomposition of a 2-edge connected graph and 3-edge connected components, enables us to achieve running time $O( m \alpha(m,n) )$.

Reducible chains of planar 1-cycle resonant graphs

A connected graph G is said to be k-cycle resonant if, for 1⩽t⩽k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and sufficient conditions for a graph to be k-cycle resonant were given. Recently, some necessary and sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant were also given. In this paper, we investigate 1-cycle-resonant reducible (simply 1-CR-reducible) chains of planar 1-cycle resonant graphs. The upper and lower bounds of the numbers of 1-CR-reducible chains of planar 1-cycle resonant graphs are given. Consequently, it is shown that any planar 1-cycle resonant graph has an ear decomposition. Furthermore, a construction method of planar 1-cycle resonant graphs and an efficient algorithm for recognizing 1-cycle resonant graphs are established.

Reduced 2-to-1 maps and decompositions of graphs with no 2-to-1 cut sets

A graph has an increasing ear decomposition if it can be constructed from a simple closed curve by attaching arcs in stages with the endpoints of each arc attached to different points so that at least one new branch point is formed at each stage. A reduced 2-to-1 map is a 2-to-1 map that does not have a restriction that is 2-to-1. A 2-to-1 cut set of a graph G is a finite subset B such that G⧹ B has at least 2| B| components. A graph has an increasing ear decomposition if and only if it does not have a 2-to-1 cut set, and a graph is the image of a reduced 2-to-1 map if and only if it does not have a 2-to-1 cut set.

Efficient Parallel Graph Algorithms for Coarse-Grained Multicomputers and BSP

In this paper we present deterministic parallel algorithms for the coarse-grained multicomputer (CGM) and bulk synchronous parallel (BSP) models for solving the following well-known graph problems: (1) list ranking, (2) Euler tour construction in a tree, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, and (7) 2-edge connectivity and biconnectivity (testing and component computation). The algorithms require O(log p) communication rounds with linear sequential work per round (p = no. processors, N = total input size). Each processor creates, during the entire algorithm, messages of total size O(log (p) (N/p)) . The algorithms assume that the local memory per processor (i.e., N/p ) is larger than p e , for some fixed e > 0 . Our results imply BSP algorithms with O(log p) supersteps, O(g log (p) (N/p)) communication time, and O(log (p) (N/p)) local computation time. It is important to observe that the number of communication rounds/ supersteps obtained in this paper is independent of the problem size, and grows only logarithmically with respect to p . With growing problem size, only the sizes of the messages grow but the total number of messages remains unchanged. Due to the considerable protocol overhead associated with each message transmission, this is an important property. The result for Problem (1) is a considerable improvement over those previously reported. The algorithms for Problems (2)—(7) are the first practically relevant parallel algorithms for these standard graph problems.

Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice

This is a sequel to our papers (M. H. de Carvalho, C. L. Lucchesi, and U. S. R. Murty, 1999, Combinatorica19, 151–174; 2001, J. Combin. Theory, Ser. B; and 2001, J. Combin. Theory, Ser. B). A Petersen brick is a graph whose underlying simple graph is isomorphic to the Petersen graph. For a matching covered graph G, b(G) denotes the number of bricks of G, and p(G) denotes the number of Petersen bricks of G. An ear decomposition of G is optimal if, among all ear decompositions of G, it uses the least possible number of double ears. Here we make use of the main theorem in (2001, J. Combin. Theory, Ser. B) to prove that the number of double ears in an optimal ear decomposition of a matching covered graph G is b(G)+p(G). In particular, if G is a brick that is not a Petersen brick, then there is an ear decomposition of G with exactly one double ear. This answers a question raised by D. Naddef and W. R. Pulleyblank (1982, Ann. Discrete Math.16, 241–260). Using this theorem, we give an alternative proof of L. Lovász' matching lattice characterization theorem (1987, J. Combin. Theory, Ser. B43, 187–222). We also show that for any matching covered graph G, there is a basis for the matching lattice of G consisting of incidence vectors of perfect matchings of G. This answers a question raised by U. S. R. Murty (1994, “The Matching Lattice and Related Topics,” Technical Report, University of Waterloo). In fact, we show that such a basis may be obtained from the incidence vectors of perfect matchings associated with optimal ear decompositions of G.

Ear Decomposition for Pair Comparison Data

An efficient graph-theoretical decomposition technique is introduced that treats inconsistencies in behavioral data as systematic adaptations rather than random errors. This technique, which is known as ear decomposition, reduces inconsistencies in any binary data set to a basis of directed cycles. Such a basis characterizes the data set in terms of inconsistencies and its size offers an improved measure of internal consistency. In two examples it is illustrated how different implementations of the ear decomposition technique can help to identify choices that are critical for violations of transitivity.

Towards a characterisation of Pfaffian near bipartite graphs

A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of K 3,3 such that G− VH contains a 1-factor. However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs is currently not known. The 2-ear theorem of Lovász and Plummer is likely to play a crucial rôle in such a characterisation. The theorem asserts that every 1-extendible graph G has an ear decomposition K 2= G 0, G 1,…, G t = G such that the 1-extendible graph G i is obtained from the 1-extendible graph G i−1 by the adjunction of one or two ears. In this paper we first show that we can restrict the study of Pfaffian graphs to 1-extendible graphs which have an ear decomposition with a unique 2-ear adjunction. Motivated by that result we start a characterisation of Pfaffian graphs with an ear decomposition where the unique 2-ear adjunction is the final adjunction, i.e., G t is obtained from G t−1 by the adjunction of two ears. Such graphs are called ‘near bipartite’.

Optimal Randomized EREW PRAM Algorithms for Finding Spanning Forests

We present the first randomized O(logn) time and O(m+n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G=(V,E) with n vertices and m edges. Our algorithm is optimal with respect to time, work, and space. As a consequence we get optimal randomized EREW PRAM algorithms for other basic connectivity problems such as finding a bipartite partition, finding bridges and biconnected components, finding Euler tours in Eulerian graphs, finding an ear decomposition, finding an open ear decomposition, finding a strong orientation, and finding an st-numbering.

Generic rigidity of molecular graphs via ear decomposition

A basic model in the study of structural rigidity is a network of rigid bars connected at their endpoints by flexible joints; this can be represented by what is called here a frame, which is a graph G realized in Euclidean space with straight-line edges. The edges act as constraints, fixing the distances between vertices. The degrees of freedom of a frame, i.e., the dimension of the space of infinitesimal motions of its vertices, is a standard measure of its flexibility. A long-standing open problem is to find a purely combinatorial algorithm for computing the generic degrees of freedom in three dimensions, ψ( G), a number which depends on the graph alone. This paper focuses on the special case of molecular graphs and their frames, in which an additional edge is added for each edge-pair angle (i.e., the squares of graphs); such graphs are used to model atom-bond networks with bond-angle forces, which arise in the study of glasses. The main result of this paper is a recurrence relation, which, when combined with previous work by the author, yields a practical algorithm to produce both upper and lower bounds on ψ( G). The result is stated for molecular graphs in three dimensions, but has analogs for standard graphs in two or three dimensions. The algorithm uses a generalization of ear decomposition, here called chain decomposition. The paper gives as a corollary a simple characterization of two classes of molecular graphs for which one can compute ψ( G) exactly. The paper also gives the following theorem: if a graph is edge two-connected and has no chordless cycle of length greater than 6, the corresponding molecular graph is rigid.

Evolutionary Families of Sets

A finite family of subsets of a finite set is said to be evolutionary if its members can be ordered so that each subset except the first has an element in the union of the previous subsets and also an element not in that union. The study of evolutionary families is motivated by a conjecture of Naddef and Pulleyblank concerning ear decompositions of 1-extendable graphs. The present paper gives some sufficient conditions for a family to be evolutionary.