Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDijkstra Graphs
285
0
0.0
(
0
)
Added by
Vinicius Gusmao Pereira de Sa
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We revisit a concept that has been central in some early stages of computer science, that of structured programming: a set of rules that an algorithm must follow in order to acquire a structure that is desirable in many aspects. While much has been written about structured programming, an important issue has been left unanswered: given an arbitrary, compiled program, describe an algorithm to decide whether or not it is structured, that is, whether it conforms to the stated principles of structured programming. We refer to the classical concept of structured programming, as described by Dijkstra. By employing a graph model and graph-theoretic techniques, we formulate an efficient algorithm for answering this question. To do so, we first introduce the class of graphs which correspond to structured programs, which we call Dijkstra Graphs. Our problem then becomes the recognition of such graphs, for which we present a greedy $O(n)$-time algorithm. Furthermore, we describe an isomorphism algorithm for Dijkstra graphs, whose complexity is also linear in the number of vertices of the graph. Both the recognition and isomorphism algorithms have potential important applications, such as in code similarity analysis.
rate research
Read More
For any class $mathcal{C}$ of bipartite graphs, we define quasi-$cal C$ to be the class of all graphs $G$ such that every bipartition of $G$ belongs to $cal C$. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.
Different graph generalizations have been recently used in an ad-hoc manner to represent multilayer networks, i.e. systems formed by distinct layers where each layer can be seen as a network. Similar constructions have also been used to represent time-varying networks. We introduce the concept of MultiAspect Graph (MAG) as a graph generalization that we prove to be isomorphic to a directed graph, and also capable of representing all previous generalizations. In our proposal, the set of vertices, layers, time instants, or any other independent features are considered as an aspect of the MAG. For instance, a MAG is able to represent multilayer or time-varying networks, while both concepts can also be combined to represent a multilayer time-varying network and even other higher-order networks. Since the MAG structure admits an arbitrary (finite) number of aspects, it hence introduces a powerful modelling abstraction for networked complex systems. This paper formalizes the concept of MAG and derives theoretical results useful in the analysis of complex networked systems modelled using the proposed MAG abstraction. We also present an overview of the MAG applicability.
A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the general recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.
A graph $G$ is said to be the intersection of graphs $G_1,G_2,ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=cdots=V(G_k)$ and $E(G)=E(G_1)cap E(G_2)capcdotscap E(G_k)$. For a graph $G$, $mathrm{dim}_{COG}(G)$ (resp. $mathrm{dim}_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $mathrm{dim}_{COG}(G)leqmathrm{tw}(G)+2$, (b) $mathrm{dim}_{TH}(G)leqmathrm{pw}(G)+1$, and (c) $mathrm{dim}_{TH}(G)leqchi(G)cdotmathrm{box}(G)$, where $mathrm{tw}(G)$, $mathrm{pw}(G)$, $chi(G)$ and $mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $mathrm{dim}_{COG}(G)$ and $mathrm{dim}_{TH}(G)$ when $G$ belongs to some special graph classes.
A $k$-frugal colouring of a graph $G$ is a proper colouring of the vertices of $G$ such that no colour appears more than $k$ times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, planar graphs with large girth, and outerplanar graphs, and relate this parameter with several well-studied colourings, such as colouring of the square, cyclic colouring, and $L(p,q)$-labelling. We also study frugal edge-colourings of multigraphs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
