Ear Decomposition Research Articles

Loose ear decompositions and their applications to right-angled Artin groups

We characterize planar graphs and graph minors among other graph theoretic notions in terms of right-angled Artin groups (RAAGs). For this, we determine all sets of elements in RAAGs with ears as underlying graphs that are exactly the sets of vertex generators. Generalizing ear decompositions of graphs to loose ear decompositions, we characterize both decompositions in terms of RAAGs. The desired results follow as applications of loose ear decompositions of RAAGs.

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

Removable edges in near‐bipartite bricks

AbstractAn edge of a matching covered graph is removable if is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than and has at least removable edges. A brick is near‐bipartite if it has a pair of edges such that is a bipartite matching covered graph. In this paper, we show that in a near‐bipartite brick with at least six vertices, every vertex of , except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, has at least removable edges. Moreover, all graphs attaining this lower bound are characterized.

Removable and forced subgraphs of graphs

A connected graph G is H-removable if H is a subgraph of G, G has at least |V(H)|+2 vertices and G−V(H′) has a perfect matching for every subgraph H′ of G isomorphic to H. So a connected graph is kK2-removable if and only if it is k-extendable, where kK2 is a matching of size k. Further G is an H-forced graph if G−V(H′) has a unique perfect matching for every subgraph H′ of G isomorphic to H. In this paper, we first characterize kK2-forced graphs for k≥2 as K2k+2 and Kk+1,k+1 by using randomly matchable graphs. Let Pi denote a path with i vertices. We show that P3- and P4-removable graphs are factor-critical and 1-extendable respectively. By using ear decomposition, we obtain that a connected graph G is P3-forced if and only if G is K5 or C2k+1 (k≥2). Finally we prove that a connected graph G is P4-forced if and only if G is isomorphic to K6, K3,3, C2k (k≥3), the Petersen graph, a uniform theta graph Θ3n (n≥3) or C12+.

Joins, ears and Castelnuovo–Mumford regularity

We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[EG] be the polynomial ring on the edges of G and K[VG] the polynomial ring on the vertices of G. We associate to G an ideal, I(XG), defined as the preimage of (xi2−xj2:i,j∈VG)⊆K[VG] by the map K[EG]→K[VG] which sends a variable, te, associated to an edge e={i,j}, to the product xixj of the variables associated to its vertices. We show that K[EG]/I(XG) is a one-dimensional, Cohen–Macaulay, graded ring and we relate its degree with the number of vertices and connected components of the graph. We show that I(XG) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo–Mumford regularity of I(XG) providing the following sharp upper and lower bounds:μ(G)≤regI(XG)≤|VG|−b0(G)+1, where μ(G) is the maximum vertex join number of the graph and b0(G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

Generating simple near‐bipartite bricks

AbstractA brick is a 3‐connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is near‐bipartite if it has a pair of edges and such that is bipartite and matching covered; examples are and the triangular prism . The significance of near‐bipartite bricks arises from the theory of ear decompositions of matching covered graphs. The object of this paper is to establish a generation procedure which is specific to the class of simple near‐bipartite bricks. In particular, we prove that every simple near‐bipartite brick has an edge such that the graph obtained from by contracting each edge that is incident with a vertex of degree two is also a simple near‐bipartite brick, unless belongs to any of eight well‐defined infinite families. This is a refinement of the brick generation theorem of Norine and Thomas which is applicable to the class of near‐bipartite bricks. Earlier, the first author proved a similar generation theorem for (not necessarily simple) near‐bipartite bricks; we deduce our main result from this theorem. Our proof is based on a strategy of Carvalho, Lucchesi and Murty and uses several of their techniques and results.

A characterization of nonfeasible sets in matching covered graphs

AbstractLet be a matching covered graph and let . Then is called feasible if there exist two perfect matchings and in such that ; otherwise, it is nonfeasible. For any vertex , the switching operation of in is defined as the symmetric difference of and , where denotes the set of all edges in incident with . Two sets are switching‐equivalent if can be obtained from by a series of switching operations and vice visa. Lukot'ka and Rollová showed that if is a regular bipartite graph, then is nonfeasible if and only if is switching‐equivalent to (as well as ). In this paper, we give a complete characterization of nonfeasible sets in matching covered graphs. We prove that if is a matching covered graph and is an arbitrary nonfeasible set of , then is switching‐equivalent to both and if and only if is bipartite, and is switching‐equivalent to either or (but not both) if and only if is nonbipartite and has an ear decomposition with exactly one double ear. We also show that for any two integers and with and , there exist infinitely many ‐regular simple graphs of class 1 with arbitrarily large brick number, connectivity and a nonfeasible set such that is not switching‐equivalent to either or . This gives negative answers to two problems proposed by Lukot'ka and Rollová and by He et al, respectively.

Irreducible decomposition of powers of edge ideals

In this paper by using some tools from graph theory, mainly the theorem on ear decomposition of factor-critical graphs given by L. Lovász and the canonical decomposition of a graph given by Edmonds and Gallai, we can describe each irreducible component of powers of edge ideals of a graph. As an application we use our results to persistence of irreducible components, to bound dstab and to count the number of irreducible components of powers of edge ideals of a graph.

On nonfeasible edge sets in matching‐covered graphs

AbstractLet be a matching‐covered graph and be an edge set of . is said to be feasible if there exist two perfect matchings and in such that . For any is said to be switching‐equivalent to , where is the set of edges in each of which has exactly one end in and is the symmetric difference of two sets and . Lukot'ka and Rollová showed that when is regular and bipartite, is nonfeasible if and only if is switching‐equivalent to . This article extends Lukot'ka and Rollová's result by showing that this conclusion holds as long as is matching‐covered and bipartite. This article also studies matching‐covered graphs whose nonfeasible edge sets are switching‐equivalent to or and partially characterizes these matching‐covered graphs in terms of their ear decompositions. Another aim of this article is to construct infinite many ‐connected and ‐regular graphs of class 1 containing nonfeasible edge sets not switching‐equivalent to either or for an arbitrary integer with , which provides a negative answer to a problem proposed by He et al.

Ear Decomposition and Balanced Neighborly Simplicial Manifolds

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.

Channel assignment problem and relaxed 2-distant coloring of graphs

Let G be a simple graph. Suppose f is a mapping from V(G) to nonnegative integers. If, for any two adjacent vertices u and v of G, |f(u)−f(v)|≥2, then f is called a 2-distant coloring of G. In this paper, we introduce a relaxation of 2-distant coloring of a graph. Let t be a nonnegative integer. Suppose f is a mapping from V(G) to nonnegative integers. If adjacent vertices receive different integers and for each vertex u of G, the number of neighbors v of u with |f(v)−f(u)|=1 is at most t, then f is called a t-relaxed 2-distant coloring of G. If t=0 then f is just a 2-distant coloring of G. The span of f, denote by sp(f), is the difference between the maximum and minimum integers used by f. The minimum span of a t-relaxed 2-distant coloring of G, is called t-relaxed 2-distant coloring span of G, denoted by sp2t(G). Suppose G is a graph. Let γ:V(G)→Z+ be a function defined on V(G). A γ-relaxed 2-distant coloring of G with span k is a mapping f from V(G) to {0,1,…,k} such that f(u)≠f(v) for any two adjacent vertices u and v and each vertex u of G has at most γ(u) neighbors v with |f(v)−f(u)|=1.In this paper, we prove that, for any two positive integers t and k with k≥2, deciding if sp2t(G)≤k for a graph G is NP-complete, except the case when t=1 and k=2 which is polynomial-time solvable. Let t be any nonnegative integer. It is easy to see that sp2t(G)≤6 for any planar graph G and sp2t(G)≤4 for any outerplanar graph G. We prove that, for any two positive integers t and k with k∈{2,3,4,5}, deciding if sp2t(G)≤k for a planar graph G is NP-complete, except the case when t=1 and k=2 which is polynomial-time solvable. We present examples to illustrate that there is no constant integer t such that every planar graph has a t-relaxed 2-distant coloring of span 5 and there is no constant integer t such that every outerplanar graph has a t-relaxed 2-distant coloring of span 3. We introduce pseudo ear decomposition and simple pseudo ear decomposition of a graph and show that a graph is outerplanar if and only if it admits a simple pseudo ear decomposition. Using this result, we show that each outerplanar graph has a γ-relaxed 2-distant coloring with span 3, where γ(v)=⌈d(v)2⌉ for each vertex v of G and the function γ is sharp in some sense, and that every triangle-free outerplanar graph has a 1-relaxed 2-distant coloring with span 3.

Constrained ear decompositions in graphs and digraphs

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovasz states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovasz, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.

Regularity of the vanishing ideal over a bipartite nested ear decomposition

We study the Castelnuovo–Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length [Formula: see text] increases by [Formula: see text], where [Formula: see text] is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, [Formula: see text], endowed with a weak nested ear decomposition is equal to [Formula: see text] where [Formula: see text] is the number of even length ears and pendant edges of the decomposition. As a corollary, we show that for bipartite graph the number of even length ears in a nested ear decomposition starting from a vertex is constant.

Associated primes of powers of edge ideals and ear decompositions of graphs

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of commutative algebra and combinatorics. It covers all previous major results in this topic and has several interesting consequences.

Tools for counting odd cycles in graphs

Let k be a positive integer at least 2. A graph is k-critical if it can be colored in no fewer than k colors and its proper subgraphs can be colored in fewer than k colors. T. Gallai asked whether a k-critical graph on n vertices contains n distinct (k−1)-critical subgraphs. This paper gives an affirmative answer to Gallai's question restricted to 4-critical graphs by proving they contain at least 83n−293 odd cycles.For 2-connected nonbipartite graphs, it is shown that the cycle space of such a graph is generated by its odd cycles and that its cyclomatic number provides a sharp lower bound for the number of its odd cycles.For 3-connected nonbipartite graphs, it is shown that the cycle space of such a graph is generated by the set of odd cycles containing any fixed vertex. An initially bipartite ear decomposition is described and is used to show that these graphs contain a spanning 2-connected bipartite subgraph, and contain at least as many odd cycles as the number of their vertices.

Ear decomposition and induced even cycles

Lovász shows that a matching covered graph G has an ear decomposition starting with an arbitrary edge of G. A 2-edge-connected cubic graph is matching covered. Using Oum’s characterization of 2-edge-connected claw-free cubic graphs (Oum, 2011) , this paper gives a characterization of 2-connected claw-free cubic graphs which have ear decompositions starting with an arbitrary induced even cycle.

Simple computation of st-edge- and st-numberings from ear decompositions

We propose simple algorithms for computing st-numberings and st-edge-numberings of graphs with running time O(m). Unlike previous serial algorithms, these are not dependent on an initially chosen DFS-tree. Instead, we compute st-(edge-)numberings that are consistent with any open ear decomposition D of a graph in the sense that every ear of D is numbered increasingly or decreasingly.Recent applications need such st-numberings, and the only two linear-time algorithms that are known for this task use a complicated order data structure as black box. We avoid using this data structure by introducing a new and light-weight numbering scheme. In addition, we greatly simplify the recent algorithms for computing (the much less known) st-edge-numberings.

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths ( APSP ) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi- and many-core architectures. It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures. In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortest-paths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.

A heuristic approach for dividing graphs into bi-connected components with a size constraint

In this paper we propose a new problem of finding the maximal bi-connected partitioning of a graph with a size constraint (MBCPG-SC). With the goal of finding approximate solutions for the MBCPG-SC, a heuristic method is developed based on the open ear decomposition of graphs. Its essential part is an adaptation of the breadth first search which makes it possible to grow bi-connected subgraphs. The proposed randomized algorithm consists of growing several subgraphs in parallel. The quality of solutions generated in this way is further improved using a local search which exploits neighboring relations between the subgraphs. In order to evaluate the performance of the method, an algorithm for generating pseudo-random unit disc graphs with known optimal solutions is created. Computational experiments have also been conducted on graphs representing electrical distribution systems for the real-world problem of dividing them into a system of fault tolerant interconnected microgrids. The experiments show that the proposed method frequently manages to find optimal solutions and has an average error of only a few percent to known optimal solutions. Further, it manages to find high quality approximate solutions for graphs having up to 10,000 nodes in reasonable time.

The joint movement of pebbles in solving the ( $$N^2-1$$ N 2 - 1 )-puzzle suboptimally and its applications in rule-based cooperative path-finding

The present paper deals with the problem of solving the ($$n^2 - 1$$n2-1)-puzzle and cooperative path-finding (CPF) problems sub-optimally by rule-based algorithms. To solve the puzzle, we need to rearrange $$n^2 - 1$$n2-1 pebbles in the $$n \times n$$n×n-sized square grid using one vacant position to achieve the goal configuration. An improvement to the existing polynomial-time algorithm is proposed and experimentally analyzed. The improved algorithm represents an attempt to move pebbles in a more efficient way compared to the original algorithm by grouping them into so-called snakes and moving them together as part of a snake formation. An experimental evaluation has shown that the snakeenhanced algorithm produces solutions which are 8---9 % shorter than the solutions generated by the original algorithm. Snake-like movement has also been integrated into the rule-based algorithms used in solving CPF problems sub-optimally, which is a closely related task. The task in CPF consists in moving a group of abstract robots on an undirected graph to specific vertices. The robots can move to unoccupied neighboring vertices; no more than one robot can be placed in each vertex. The ($$n^2 - 1$$n2-1)-puzzle is a special case of CPF where the underlying graph is a 4-connected grid and only one vertex is vacant. Two major rule-based algorithms for solving CPF problems were included in our study--BIBOX and PUSH-and-SWAP (PUSH-and-ROTATE). The use of snakes in the BIBOX algorithm led to consistent efficiency gains of around 30 % for the ($$n^2 - 1$$n2-1)-puzzle and up to 50 % in for CPF problems on biconnected graphs with various ear decompositions and multiple vacant vertices. For the PUSH-and-SWAP algorithm, the efficiency gain achieved from the use of snakes was around 5---8 %. However, the efficiency gain was unstable and hardly predictable for PUSH-and-SWAP.