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We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions…
Graphons $W$ can be used as stochastic models to sample graphs $G_n$ on $n$ nodes for $n$ arbitrarily large. A graphon $W$ is said to have the $H$-property if $G_n$ admits a decomposition into disjoint cycles with probability one as $n$…
The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix C_{nm} of simple digraph and an incidence matrix F of balanced bipartite…
Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by Jeff Erickson. This problem has various applications in computational geometry, like robot motion planning, generating polygon etc. We will…
The Hamiltonian cycle problem (HCP), which is an NP-complete problem, consists of having a graph G with n nodes and m edges and finding the path that connects each node exactly once. In this paper we compare some algorithms to solve a…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular,…
In this paper, we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of $K_{1,4}$-free split graphs…
We study the Hamilton cycle problem with input a random graph G=G(n,p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In…
We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two $n \times n$ polar decompositions…
In this article, we consider nonlinear complementarity problem. We introduce a new homotopy function for finding the solution of nonlinear complementarity problem through the trajectory . We show that the homotopy path approaching the…
We state a combinatorial optimization problem whose feasible solutions define both a decomposition and a node labeling of a given graph. This problem offers a common mathematical abstraction of seemingly unrelated computer vision tasks,…
In the Online Delayed Connected H-Node-Deletion Problem, an unweighted graph is revealed vertex by vertex and it must remain free of any induced copies of a specific connected induced forbidden subgraph H at each point in time. To achieve…
Let $\mathcal{G}(k)$ denote the set of connected $k$-regular graphs $G$, $k\geq2$, where the number of vertices at distance 2 from any vertex in $G$ does not exceed $k$. Asratian (2006) showed (using other terminology) that a graph…
In 2007, Arkin et al. initiated a systematic study of the complexity of the Hamiltonian cycle problem on square, triangular, or hexagonal grid graphs, restricted to polygonal, thin, superthin, degree-bounded, or solid grid graphs. They…
${ NP}$-complete problem "Hamiltonian cycle"\ for graph $G=(V,E)$ is extended to the "Hamiltonian Complement of the Graph"\ problem of finding the minimal cardinality set $H$ containing additional edges so that graph $G=(V,E\cup H)$ is…
Nucleus decompositions have been shown to be a useful tool for finding dense subgraphs. The coreness value of a clique represents its density based on the number of other cliques it is adjacent to. One useful output of nucleus decomposition…
Every semicomplete multipartite digraph contains a quasi-Hamiltonian path, but the problem of finding a quasi-Hamiltonian path with prescribed start and end vertex is NP-complete even when restricted to semicomplete multipartite digraphs…
In this article, we investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface…
A connected graph is 4-connected if it contains at least five vertices and removing any three of them does not disconnect it. A frequent preprocessing step in graph drawing is to decompose a plane graph into its 4-connected components and…