Graph Theory Parameters Research Articles

Domination and total domination in cubic graphs of large girth

The domination number γ(G) and the total domination number γt(G) of a graph G without an isolated vertex are among the most well-studied parameters in graph theory. While the inequality γt(G)≤2γ(G) is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow us to improve the inequality by much.In the present paper we consider the relation of γ(G) and γt(G) for cubic graphs G of large girth. Clearly, in this case γ(G) is at least n(G)/4 where n(G) is the order of G. If γ(G) is close to n(G)/4, then this forces a certain structure within G. We exploit this structure and prove an upper bound on γt(G), which depends on the value of γ(G). As a consequence, we can considerably improve the inequality γt(G)≤2γ(G) for cubic graphs of large girth.

Graph Theory Approach to Transportation Systems Design and Optimization

The main aim of the paper is to present graph theory parameters and algorithms as tool to analyze and to optimise transportation systems. To realize these goals the 0-1 knapsack problem solution by SPEA algorithm, methods and procedures for finding the minimal spanning tree in graphs and digraphs, domination parameters problems accurate to analyse the transportation systems are introduced and described. Possibility of application of graph theory algorithms and parameters to analyze exemplary transportation system are shown.

Some Defective Parameters in Graphs

Given a graph G and an integer k, a set S of vertices in G is k-sparse if S induces a graph with a maximum degree of at most k. Many parameters in graph theory are defined in terms of independent sets. Accordingly, their definitions can be expanded taking into account the notion of k-sparse sets. In this discussion, we examine several of those extensions. Similarly, S is k-dense if S induces a k-sparse graph in the complement of G. A partition of V(G) is a k-defective cocoloring if each part is k-sparse or k-dense. The minimum order of all k-defective cocolorings is the k-defective cochromatic number of G and denoted z k (G). Analogous notions are defined similarly for k-defective coloring where V(G) is partitioned into k-sparse parts. We show the NP-hardness of computing maximum k-defective sets in planar graphs with maximum degree at most k + 1 and arbitrarily large girth. We explore the extension of Ramsey numbers to k-sparse and k-dense sets of vertices. Lastly, we discuss some bounds related to k-defective colorings and k-defective cocolorings.

Searching for a Visible, Lazy Fugitive

Graph searching problems are described as games played on graphs, between a set of cops and a fugitive. Variants of the game restrict the abilities of the cops and the fugitive and the corresponding search numbers (the least number of cops that have a winning strategy) are related to several well-known parameters in graph theory. We study the case where the fugitive is visible (the cops’ strategy can take into account his current position) and lazy (he moves only when the cops move to his position). Our results are stated and proven in a general setting where the fugitive’s speed (i.e., the lengths of paths he can move along) can be unbounded or bounded by some constant. We give a min-max characterization of the corresponding parameters, which we show to be computable in polynomial time for fugitives with unbounded speed and speed at most 3 and to be nondeterministic polynomial-time complete (NP-complete) for all other finite speeds. This is in contrast to the other standard versions of the game, where the parameters corresponding to fugitives with unbounded speed are NP-complete. Several consequences of our results are also discussed.

Working memory in aging. Graph theory for a MEG-DTI study

Event Abstract Back to Event Working memory in aging. Graph theory for a MEG-DTI study Jose Angel Pineda Pardo1*, Julia Guiomar Niso Galán1, Ricardo Gutierrez Díez1, Elena Solesio1, Fernando Maestú Unturbe1, 2 and Francisco Del Pozo Guerrero1 1 Center for Biomedical Technology, Universidad Politécnica de Madrid, Spain 2 Dept. of Psicología Básica II, Universidad Complutense de Madrid, Spain The study of age-related changes in phase synchronization on MEG recordings of a visual-auditory delayed match-to-sample task with interference provides interesting clues of how memory-related brain connectivity patterns arise and how they are sustained during the presentation of interfering information. In this work we integrated the synchronization analysis with the fractional anisotropy images using graph theory. Participants: twelve elderly healthy participants. Task: visual-auditory delayed match-to-sample task in two conditions: active and passive interference for MEG data. Additionally, a MR-DWI was acquired for each subject. The study is focused on the comparison of the 500ms prior to the interference and the 500ms after the 3 seconds-interference ends. Synchronization analysis. Phase Locking Values were computed for 50ms-time windows in the different frequency bands with the MEG data. A cluster-based permutation test based was performed. Complex network analysis: based on the synchronization values, graph theory parameters, such as mean shortest path, node outreach or network modularity, were obtained to characterize the functional brain networks involved. Multiple regression analysis: the previous results were correlated with participants’ fractional anisotropy images. We expect to observe significant differences between active and passive condition, which should be present not only in the synchronization analysis, but also reflected in the graph analysis results and correlated with the MRI-DWI in specific brain regions. Conclusions: MEG synchronization analysis combined with MRI-DWI offers multimodal information able to characterize brain networks underlying human cognition during the performance of a concrete task. Keywords: Aging, MEG Conference: XI International Conference on Cognitive Neuroscience (ICON XI), Palma, Mallorca, Spain, 25 Sep - 29 Sep, 2011. Presentation Type: Poster Presentation Topic: Poster Sessions: Cognitive Aging Citation: Pineda Pardo J, Niso Galán J, Gutierrez Díez R, Solesio E, Maestú Unturbe F and Del Pozo Guerrero F (2011). Working memory in aging. Graph theory for a MEG-DTI study. Conference Abstract: XI International Conference on Cognitive Neuroscience (ICON XI). doi: 10.3389/conf.fnhum.2011.207.00608 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 15 Nov 2011; Published Online: 28 Nov 2011. * Correspondence: Dr. Jose Angel Pineda Pardo, Center for Biomedical Technology, Universidad Politécnica de Madrid, Madrid, Spain, jappxiclana85@gmail.com Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jose Angel Pineda Pardo Julia Guiomar Niso Galán Ricardo Gutierrez Díez Elena Solesio Fernando Maestú Unturbe Francisco Del Pozo Guerrero Google Jose Angel Pineda Pardo Julia Guiomar Niso Galán Ricardo Gutierrez Díez Elena Solesio Fernando Maestú Unturbe Francisco Del Pozo Guerrero Google Scholar Jose Angel Pineda Pardo Julia Guiomar Niso Galán Ricardo Gutierrez Díez Elena Solesio Fernando Maestú Unturbe Francisco Del Pozo Guerrero PubMed Jose Angel Pineda Pardo Julia Guiomar Niso Galán Ricardo Gutierrez Díez Elena Solesio Fernando Maestú Unturbe Francisco Del Pozo Guerrero Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs

Consider a connected, undirected graph $G=(V,E)$ and an integer $r \geq 1$; for any vertex $v\in V$, let $B_r(v)$ denote the ball of radius $r$ centred at $v$, i.e., the set of all vertices linked to $v$ by a path consisting of at most $r$ edges. If for all vertices $v \in V$, the sets $B_r(v)$ are different, then we say that $G$ is $r$-twin-free.In $r$-twin-free graphs, we prolong the study of the extremal values that can be achieved by the main classical parameters in graph theory, and investigate here the number of edges, the minimum degree, the size of a maximum independent set, as well as radius and diameter.

Wigner-Racah Algebra Approach to Caselle-PonzanoFusion Rules

Caselle-Ponzano fusion rules, based on graph theory, are investigated in the framework of Wigner-Racah algebras. While in the graph theory parameters of fusion rules are taken over specific values, the restriction of parameters is removed in the present formalism. Formal extension to q -analogs is also given.