A Survey of Best Monotone Degree Conditions for Graph Properties

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvatal's well-known degree condition for hamiltonicity is best possible.Comment: 25 page

An Efficient Algorithm to Test Forcibly-connectedness of Graphical Degree Sequences

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al\u27s classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage\u27s classic algorithm to enumerate graphical partitions of even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length $n$ and forcibly connected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) almost all zero-free graphical degree sequences of length $n$ are forcibly connected; (2) almost none of the graphical partitions of even $n$ are forcibly connected

An efficient algorithm to test forcibly-connectedness of graphical degree sequences

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al's classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage's classic algorithm to enumerate graphical partitions of even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length $n$ and forcibly connected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) almost all zero-free graphical degree sequences of length $n$ are forcibly connected; (2) almost none of the graphical partitions of even $n$ are forcibly connected.Comment: 20 pages, 11 table

Forcibly-biconnected Graphical Degree Sequences: Decision Algorithms and Enumerative Results

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length $n$ and forcibly biconnected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) the proportion of forcibly biconnected graphical degree sequences of length $n$ among all zero-free graphical degree sequences of length $n$ is asymptotically a constant $C$ ($

Rao's Theorem for forcibly planar sequences revisited

We consider the graph degree sequences such that every realisation is a polyhedron. It turns out that there are exactly eight of them. All of these are unigraphic, in the sense that each is realised by exactly one polyhedron. This is a revisitation of a Theorem of Rao about sequences that are realised by only planar graphs. Our proof yields additional geometrical insight on this problem. Moreover, our proof is constructive: for each graph degree sequence that is not forcibly polyhedral, we construct a non-polyhedral realisation

On the Use of Graphs for Node Connectivity in Wireless Sensor Networks for Hostile Environments

[EN] Wireless sensor networks (WSNs) have been extensively studied in the literature. However, in hostile environments where node connectivity is severely compromised, the system performance can be greatly affected. In this work, we consider such a hostile environment where sensor nodes cannot directly communicate to some neighboring nodes. Building on this, we propose a distributed data gathering scheme where data packets are stored in different nodes throughout the network instead to considering a single sink node. As such, if nodes are destroyed or damaged, some information can still be retrieved. To evaluate the performance of the system, we consider the properties of different graphs that describe the connections among nodes. It is shown that the degree distribution of the graph has an important impact on the performance of the system. A teletraffic analysis is developed to study the average buffer size and average packet delay. To this end, we propose a reference node approach, which entails an approximation for the mathematical modeling of these networks that effectively simplifies the analysis and approximates the overall performance of the system.The authors wish to thank the Consejo Nacional de Ciencia y Tecnologia (CONACyT), the Comision de Operacion y Fomento de Actividades Academicas, Instituto Politecnico Nacional (COFAA-IPN, project numbers 20196225 and 20196678), and the Estimulos al Desempeno de los Investigadores del Instituto Politecnico Nacional (EDI-IPN) for the support given for this work. The work of V. Pla was supported by Grant PGC2018-094151-B-I00 (MCIU/AEI/FEDER, UE).García-González, E.; Chimal-Eguía, JC.; Rivero-Angeles, ME.; Pla, V. (2019). On the Use of Graphs for Node Connectivity in Wireless Sensor Networks for Hostile Environments. Journal of Sensors. 2019:1-22. https://doi.org/10.1155/2019/7409329S1222019Eren, T. (2017). The effects of random geometric graph structure and clustering on localizability of sensor networks. International Journal of Distributed Sensor Networks, 13(12), 155014771774889. doi:10.1177/1550147717748898Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661-703. doi:10.1137/070710111Hakimi, S. L. (1962). On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph. I. Journal of the Society for Industrial and Applied Mathematics, 10(3), 496-506. doi:10.1137/011003