On universally rigid frameworks on the line

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

Restricted t-matchings in bipartite graphs

A graph G is said to be a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph G and a subset of vertices \(R\subseteq V(G)\), the objective is to find a minimum cardinality set \(S\subset V (G)\) such that the set \(R \cup S\) induces a connected subgraph. In this paper, we present an interesting dichotomy result for STREE on bisplit graphs, we show that STREE is polynomial-time solvable for chordal bipartite bisplit graphs, and NP-complete otherwise. Further, we show that for chordal bisplit graphs, the problem is polynomial-time solvable. A revisit of our NP-complete reduction instances reveals that the instances are diameter at most 5 bipartite graphs. We also obtain one more dichotomy result for STREE on bisplit graphs which says that for diameter 5 the problem is NP-complete and polynomial-time solvable for diameter at most 4. On the parameterized complexity front, we show that the parameterized version of Steiner tree problem on bisplit graphs is fixed-parameter tractable when the parameter is the biclique size and is W[2]-hard on bisplit graphs if the parameter is the solution size. We conclude this paper by presenting structural results of bisplit graphs which will be of use to solve other combinatorial problems.KeywordsBisplit graphsThe Steiner tree problemW-hardnessDichotomy results

There are several known results concerning how matroids can be induced from given matroids by a bipartite graph and the properties that are inherited in this way. The purpose of this note is to extend some of these results to the situation where the bipartite graph is replaced by an arbitrary directed graph. We show how a directed graph and a matroid can be used to induce a new matroid. If the initial matroid is strongly base orderable, we prove that the induced matroid is also. In particular, a matroid induced from a free matroid by a directed graph is strongly base orderable. A consequence is that the cycle matroid of the complete graph on four nodes cannot be induced from a free matroid by any directed graph.

Axiomatic characterization of the interval function of a bipartite graph

Characterization of general position sets and its applications to cographs and bipartite graphs

R. Haggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K6n,6n. In [1] Cichacz and Froncek established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into generalized prisms of order 2n. In [2] and [3] Cichacz, Froncek, and Kovar showed decompositions of K3n/2,3n/2 into generalized prisms of order 2n. In this paper we present another class of complete bipartite graphs that is decomposable into prisms.

We consider the zeta function associated with the time evolution matrix of a bipartite walk on a bipartite graph and derive a formula for its absolute zeta function. Furthermore, we give formulas for the absolute zeta functions of the zeta functions of the time evolution matrices of bipartite walks on the complete bipartite graph and an even cycle graph. As an application, we obtain formulas for the absolute zeta functions of the zeta functions of the Grover matrices of the complete bipartite graph and an even cycle graph.

For a simple bipartite graph and an integer t ≥ 2, we consider the problem of finding a minimum-weight kt, t-free t-factor, which is a t-factor containing no complete bipartite graph kt, t as a subgraph. When t = 2, this problem amounts to the square-free 2-factor problem in a bipartite graph. For the unweighted square-free 2-factor problem, a combinatorial algorithm is given by Hartvigsen, and the weighted version of the problem is NP-hard. For general t, Pap designed a combinatorial algorithm for the unweighted version, and Makai gave a dual integral description of kt, t-free t-matchings for a certain case where the weight vector is vertex-induced on any subgraph isomorphic to kt, t. For this class of weight vectors, we propose a strongly polynomial algorithm to find a minimum-weight kt, t-free t-factor. The algorithm adapts the unweighted algorithms of Hartvigsen and Pap and a primal-dual approach to the minimum-cost flow problem. The algorithm is fully combinatorial and thus provides a dual integrality theorem, which is tantamount to Makai's one.

If G is a bipartite graph with bipartition A, B then let Gm,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a1, …, am, each vertex b of B by an independent set b1,…, bn, and each edge ab of G by the complete bipartite graph with edges aibj (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of Gm,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess‐current graphs.” They include new genus embeddings for a large class of bipartite graphs.

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterizations are established for the rigidity of generic frameworks for d=1 with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterization of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in one and two dimensions and to some extent between rigidity in two and three dimensions. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.

Coverings of the complete (di-)graph with n vertices by complete bipartite (di-)graphs with n vertices I

In this paper, we propose a new iterative randomized rounding algorithm for approximately solving the k-clustering minimum completion problem. The problem is defined in the telecommunication field where an instance of the problem is characterized by a bipartite graph G(S, C, E). S represents a set of services, C denotes a set of clients and E is a set related to the links between elements of C and S. The problem is to partition the initial graph into k-clusters such that each cluster is a complete bipartite subgraph. The goal is to minimize added edges to make a complete sub bipartite graph. The proposed algorithm is based upon two phases: (i) rounding a series of fractional variables, augmented with (ii) a constructive hill-climbing procedure. Both phases are embedded into an iterative search forming an iterative randomized search. The performance of the proposed method is evaluated on a set of benchmark instances taken from the literature. Provided bounds are compared to those reached by the best methods available in the literature. Encouraging results have been provided.KeywordsOptimizationGraphRounding methodTelecommunication

A (3,2)-track layout of a graph G consists of a 2-track assignment of G and an edge 3-coloring of G with no monochromatic X-crossing. This paper studies the problem of (3,2)-track layout of bipartite graph subdivisions. Recently Dujmović and Wood showed that every graph G with n vertices has a (3,2)-track subdivision of G with 4 ⌈logqn(G) ⌉ + 3 division vertices per edge, where qn(G) is the queue number of G. This paper improves their result for the case of complete bipartite graphs, and shows that every complete bipartite graph K m,n has a (3,2)-track subdivision of K m,n with 2 ⌈logqn(K m,n) ⌉ + 1 division vertices per edge, where m and n are numbers of vertices of the partite sets of K m,n with m ≥ n.