On dominator colorings in graphs

The article of record as published may be located at http://gtn.kazlow.info/GTN54.pdfGraph Theory Notes of New York LII, (2007) 25-30Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between dominator chromatic number, chromatic number, and domination number

Alliance Partition Number in Graphs

Ars Combinatoria, 103 (2012), pp. 519-529 (accepted 2007)

Degree Ranking Using Local Information

Most real world dynamic networks are evolved very fast with time. It is not feasible to collect the entire network at any given time to study its characteristics. This creates the need to propose local algorithms to study various properties of the network. In the present work, we estimate degree rank of a node without having the entire network. The proposed methods are based on the power law degree distribution characteristic or sampling techniques. The proposed methods are simulated on synthetic networks, as well as on real world social networks. The efficiency of the proposed methods is evaluated using absolute and weighted error functions. Results show that the degree rank of a node can be estimated with high accuracy using only $1\%$ samples of the network size. The accuracy of the estimation decreases from high ranked to low ranked nodes. We further extend the proposed methods for random networks and validate their efficiency on synthetic random networks, that are generated using Erd\H{o}s-R\'{e}nyi model. Results show that the proposed methods can be efficiently used for random networks as well

A Faster Method to Estimate Closeness Centrality Ranking

Closeness centrality is one way of measuring how central a node is in the given network. The closeness centrality measure assigns a centrality value to each node based on its accessibility to the whole network. In real life applications, we are mainly interested in ranking nodes based on their centrality values. The classical method to compute the rank of a node first computes the closeness centrality of all nodes and then compares them to get its rank. Its time complexity is $O(n \cdot m + n)$, where $n$ represents total number of nodes, and $m$ represents total number of edges in the network. In the present work, we propose a heuristic method to fast estimate the closeness rank of a node in $O(\alpha \cdot m)$ time complexity, where $\alpha = 3$. We also propose an extended improved method using uniform sampling technique. This method better estimates the rank and it has the time complexity $O(\alpha \cdot m)$, where $\alpha \approx 10-100$. This is an excellent improvement over the classical centrality ranking method. The efficiency of the proposed methods is verified on real world scale-free social networks using absolute and weighted error functions

Faces of NPS: Ralucca Gera, PhD

Faces of NPS features Interviews spotlighting the students, faculty, staff and alumni of our Nation’s premier defense education and research institution

Creating and understanding email communication networks to aid digital forensic investigations

Digital forensic analysts depend on the ability to understand the social networks of the individuals they investigate. We develop a novel method for automatically constructing these networks from collected hard drives. We accomplish this by scanning the raw storage media for email addresses, constructing co-reference networks based on the proximity of email addresses to each other, then selecting connected components that correspond to real communication networks. We validate our analysis against a tagged data-set of networks for which we determined ground truth through interviews with the drive owners. In the resulting social networks, we find that classical measures of centrality and community detection algorithms are effective for identifying important nodes and close associates

On the hardness of recognizing triangular line graphs

Given a graph G, its triangular line graph is the graph T(G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and link graphs. While closely related to line graphs, triangular line graphs have been difficult to understand and characterize. Van Bang Le asked if recognizing triangular line graphs has an efficient algorithm or is computationally complex. We answer this question by proving that the complexity of recognizing triangular line graphs is NP-complete via a reduction from 3-SAT.Comment: 18 pages, 8 figures, 4 table

Robustness in Nonorthogonal Multiple Access 5G Networks

The diversity of fifth generation (5G) network use cases, multiple access technologies, and network deployments requires measures of network robustness that complement throughput-centric error rates. In this paper, we investigate robustness in nonorthogonal multiple access (NOMA) 5G networks through temporal network theory. We develop a graph model and analytical framework to characterize time-varying network connectedness as a function of NOMA overloading. We extend our analysis to derive lower bounds and probability expressions for the number of medium access control frames required to achieve pairwise connectivity between all network devices. We support our analytical results through simulation

Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure