Directed Line Graphs Research Articles

Beyond Quasi-Adjoint Graphs: On Polynomial-Time Solvable Cases of the Hamiltonian Cycle and Path Problems

The Hamiltonian cycle and path problems are fundamental in graph theory and useful in modelling real-life problems. Research in this area is directed toward designing better and better algorithms for general problems, but also toward defining new special cases for which exact polynomial-time algorithms exist. In the paper, such new classes of digraphs are proposed. The classes include, among others, quasi-adjoint graphs, which are a superclass of adjoints, directed line graphs, and graphs modelling a DNA sequencing problem.

Enumeration of spanning trees of middle digraphs

Let D be a connected weighted digraph. The relation between the vertex- weighted complexity (with a fixed root) of the line digraph of D and the edge-weighted complexity (with a fixed root) of D has been given in Levine [Sandpile groups and spanning trees of directed line graphs. J Combin Theory Ser A. 2011;118:350–364] and, independently, in Sato [New proofs for Levine's theorems. Linear Algebra Appl. 2011;435:943–952]. In this paper, we obtain a relation between the vertex-weighted complexity of the middle digraph of D and the edge-weighted complexity of D. Particularly when the weight of each arc and each vertex of D is 1, the enumerative formula of spanning trees of the middle digraph of a general digraph is obtained.

Depth-based hypergraph complexity traces from directed line graphs

In this paper, we aim to characterize the structure of hypergraphs in terms of structural complexity measure. Measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders. We thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis. To locate the dominant substructure within a line graph, we identify a centroid vertex by computing the minimum variance of its shortest path lengths. A family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex. We compute the depth-based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs. The resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs. We perform (hyper)graph classification in the principal component space of the complexity trace vectors. Experiments on (hyper)graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces.

Reduced-by-matching Graphs: Toward Simplifying Hamiltonian Circuit Problem

The results presented in the paper are threefold. Firstly, a new class of reduced-by-matching directed graphs is defined and its properties studied. The graphs are output from the algorithm which, for a given 1-graph, removes arcs which are unnecessary from the point of view of searching for a Hamiltonian circuit. In the best case, the graph is reduced to a quasi-adjoint graph, what results in polynomial-time solution of the Hamiltonian circuit problem. Secondly, the systematization of several classes of digraphs, known from the literature and referring to directed line graphs, is provided together with the proof of its correctness. Finally, computational experiments are presented in order to verify the effectiveness of the reduction algorithm.

Sandpile groups and spanning trees of directed line graphs

We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph L G . The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of L G by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.

A characterization of partial directed line graphs

Can a directed graph be completed to a directed line graph? If possible, how many arcs must be added? In this paper we address the above questions characterizing partial directed line (PDL) graphs, i.e., partial subgraph of directed line graphs. We show that for such class of graphs a forbidden configuration criterion and a Krausz's like theorem are equivalent characterizations. Furthermore, the latter leads to a recognition algorithm that requires O ( m ) worst case time, where m is the number of arcs in the graph. Given a partial line digraph, our characterization allows us to find a minimum completion to a directed line graph within the same time bound. The class of PDL graphs properly contains the class of directed line graphs, characterized in [J. Blazewicz, A. Hertz, D. Kobler, D. de Werra, On some properties of DNA graphs, Discrete Appl. Math. 98(1–2) (1999) 1–19], hence our results generalize those already known for directed line graphs. In the undirected case, we show that finding a minimum line graph edge completion is NP-hard, while the problem of deciding whether or not an undirected graph is a partial graph of a simple line graph is trivial.

On fault-tolerant embedding of Hamiltonian circuits in line digraph interconnection networks

Interconnection networks based on directed line graphs have a nice property: an Euler tour in the digraph G translates to a Hamiltonian circuit in the line digraph L( G). Moreover, the existence of a Hamiltonian circuit in L( G) implies that G must be Eulerian. While Hamiltonian circuits are hard to find in the presence of link failures, the equivalent Euler tours are relatively easy to find in the presence of forbidden link pairs (corresponding to the link failures in L( G)). We explore this idea and present necessary and sufficient conditions for the existence of fault-free Euler tour in a digraph. We also present an algorithm to find such an Euler tour when it exists.

Repeated communication and Ramsey graphs

Studies the savings afforded by repeated use in two zero-error communication problems. 1. Channel coding: proving a correspondence between Ramsey numbers and independence numbers of normal graph powers, the authors show that some channels can communicate exponentially more bits in two uses than they can in one use, and that this is essentially the largest possible increase. Using probabilistic constructions of self-complementary Ramsey graphs, the authors show that similar results hold even when the number of transmissible bits is large relative to the channel's size. 2. Dual-source coding: using probabilistic colorings of directed line graphs, the authors show that there are dual sources where communicating one instance requires arbitrarily many bits, yet communicating many instances requires at most two bits per instance. For dual sources where the number of bits required for a single instance is comparable to the source's size, they employ probabilistic constructions of self-complementary Ramsey graphs that are also Cayley graphs to show that conveying two instances may require only a logarithmic number of additional bits over that needed to convey one instance. >

GIT—a heuristic program for testing pairs of directed line graphs for isomorphism

Given a pair of directed line graphs, the problem of ascertaining whether or not they are isomorphic is one for which no efficient algorithmic solution is known. Since a straightforward enumerative algorithm might require 40 years of running time on a very high speed computer in order to compare two 15-node graphs, a more sophisticated approach seems called for. The situation is similar to that prevailing in areas such as game-playing and theorem-proving, where practical algorithms are unknown (for the interesting cases), but where various practical though only partially successful techniques are available. GIT—Graph Isomorphism Tester—incorporates a variety of processes that attempt to narrow down the search for an isomorphism, or to demonstrate that none exists. No one scheme is relied upon exclusively for a solution, and the program is designed to avoid excessive computation along fruitless lines. GIT has been written in the COMIT language and successfully tested on the IBM 7090.