Neighborhood unions and hamiltonian properties in graphs

Spectra of quasi-strongly regular graphs

The so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

A proximity drawing of a graph is one in which pairs of adjacent vertices are drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called β-drawings. These drawings include as special cases the well-known Gabriel drawings (when β=1), and relative neighborhood drawings (when β=2). We first show that all biconnected outerplanar graphs are β-drawable for all values of β such that 1 ≤ β ≤ 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [15, 17], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any convex β-drawing for 1 ≤ β ≤ 2. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in a β-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected β-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.

A digraph is semicomplete if it has no pair of non-adjacent vertices. It is complete if every pair of distinct vertices induces a 2-cycle. A digraph is semicomplete multipartite if it can be obtained from a semicomplete digraph D by choosing a collection of vertex-disjoint subsets X₁,…,X_c of V(D) and then deleting all arcs both of whose end-vertices lie inside some X_i. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets V₁,…,V_c by adding the arcs of a semicomplete digraph D_i on V_i for each partite set V_i. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph S which is obtained as above from a semicomplete multipartite digraph D and semicomplete digraphs D_i=(V_i,A_i), i∈[c] such that the path or cycle uses as few arcs of A₁∪…A_c as possible. We obtain a number of results for the case when each D_i is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the D_i’s as possible in polynomial time and obtain a number of results, both structural and algorithmic on hamiltonian cycles that use the minimum or close to the minimum number of arcs from the D_i’s. Our results imply the polynomial solvability of some special cases of the NP-complete {0,1}-TSP problem. Finally we show that two natural questions about properties of quasi-hamiltonian cycles, that is, cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets.We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.

Local properties of geometric graphs

Fault-tolerant cycles embedding in hypercubes with faulty edges

Disjoint cycles covering specified vertices in bipartite graphs with partial degrees

We describe a new type of sufficient condition for a digraph to be Hamiltonian. Conditions of this type combine local structure of the digraph with conditions on the degrees of nonadjacent vertices. The main difference from earlier conditions is that we do not require a degree condition on all pairs of nonadjacent vertices. Our results generalize the classical conditions by Ghouila-Houri and Woodall. © 1996 John Wiley & Sons, Inc.

On [formula omitted]-span and [formula omitted]-span of trees and full binary trees

The sum of distances between vertices of a tree has been considered from many aspects. The question of characterizing the extremal trees that maximize or minimize various such “distance-based” graph invariants has been extensively studied. Such invariants include, to name a few, the sum of distances between all pairs of vertices and the sum of distances between all pairs of leaves. With respect to the distances between internal vertices, we provide analogous results that characterize the extremal trees that minimize the value of any nonnegative and nondecreasing function of internal distances among trees with various constraints.

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Forwarding and optical indices of 4-regular circulant networks

The first and second Zagreb indices of a graph are equal‎, ‎respectively‎, ‎to‎ ‎the sum of squares of the vertex degrees‎, ‎and the sum of the products of‎ ‎the degrees of pairs of adjacent vertices‎. ‎We now consider analogous‎ ‎graph invariants‎, ‎based on the second degrees of vertices (number of‎ ‎their second neighbors)‎, ‎called leap Zagreb indices‎. ‎A number of their‎ ‎basic properties is established‎.

A new sufficient condition for a digraph to be Hamiltonian