arXiv subjects

Dieter Mitsche

Publications and source records attributed to Dieter Mitsche.

61 records · Page 4Linked to original sources

arXiv2013

On the Fiedler value of large planar graphs

The Fiedler value $λ_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $λ_{2\max}$, and we show the bounds $2+Θ(\frac{1}{n^2}) \leq λ_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $λ_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.

math.CO↗

arXiv2013

Maximum degree in minor-closed classes of graphs

Given a class of graphs G closed under taking minors, we study the maximum degree Δ_n of random graphs from G with n vertices. We prove several lower and upper bounds that hold with high probability. Among other results, we find classes of graphs providing orders of magnitude for Δ_n not observed before, such us \log n/ \log \log \log n and \log n/ \log \log \log \log n.

math.CO↗

arXiv2012

On the Maximum Density of Graphs with Good Edge-Labellings

A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on n vertices that admits a good edge-labelling has at most n log_2(n)/2 edges, and that this bound is tight for infinitely many values of n. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every n we also construct an n-vertex graph that admits a good edge-labelling and has n log_2(n)/2 - O(n) edges.

math.CO↗

arXiv2012

Social-Aware Forwarding Improves Routing Performance in Pocket Switched Networks

Several social-aware forwarding strategies have been recently introduced in opportunistic networks, and proved effective in considerably in- creasing routing performance through extensive simulation studies based on real-world data. However, this performance improvement comes at the expense of storing a considerable amount of state information (e.g, history of past encounters) at the nodes. Hence, whether the benefits on routing performance comes directly from the social-aware forwarding mechanism, or indirectly by the fact state information is exploited is not clear. Thus, the question of whether social-aware forwarding by itself is effective in improving opportunistic network routing performance remained unaddressed so far. In this paper, we give a first, positive answer to the above question, by investigating the expected message delivery time as the size of the net- work grows larger.

cs.NI↗

arXiv2012

Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games.

cs.DM↗

arXiv2010

On the Number of Higher Order Delaunay Triangulations

Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-$k$ Delaunay if the circumcircle of each triangle of the triangulation contains at most $k$ points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is $2^{n-3}$. Next we show that uniformly distributed points have an expected number of at least $2^{ρ_1 n(1+o(1))}$ first order Delaunay triangulations, where $ρ_1$ is an analytically defined constant ($ρ_1 \approx 0.525785$), and for $k > 1$, the expected number of order-$k$ Delaunay triangulations (which are not order-$i$ for any $i < k$) is at least $2^{ρ_k n(1+o(1))}$, where $ρ_k$ can be calculated numerically.

cs.CG↗

arXiv2007

Dynamic Random Geometric Graphs

In this work we introduce Dynamic Random Geometric Graphs as a basic rough model for mobile wireless sensor networks, where communication distances are set to the known threshold for connectivity of static random geometric graphs. We provide precise asymptotic results for the expected length of the connectivity and disconnectivity periods of the network. We believe the formal tools developed in this work could be of use in future studies in more concrete settings. In addition, for static random geometric graphs at the threshold for connectivity, we provide asymptotic expressions on the probability of existence of components according to their sizes.

cs.DM↗