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

SOME RESULTS OF LABELING ON BROOM GRAPH

A Broom Graph Bn,d  is a graph of n vertices, which have a path P with d vertices and (n-d) pendant vertices, all of these being adjacent to either the origin u or the terminus v of the path P. Here we consider various labeling on Broom graph such as Cordial labeling, Antimagic labeling and b-coloring. &nbsp

Textual data summarization using the Self-Organized Co-Clustering model

International audienceRecently, different studies have demonstrated the use of co-clustering, a data mining technique which simultaneously produces row-clusters of observations and column-clusters of features. The present work introduces a novel co-clustering model to easily summarize textual data in a document-term format. In addition to highlighting homogeneous co-clusters as other existing algorithms do we also distinguish noisy co-clusters from significant co-clusters, which is particularly useful for sparse document-term matrices. Furthermore, our model proposes a structure among the significant co-clusters, thus providing improved interpretability to users. The approach proposed contends with state-of-the-art methods for document and term clustering and offers user-friendly results. The model relies on the Poisson distribution and on a constrained version of the Latent Block Model, which is a probabilistic approach for co-clustering. A Stochastic Expectation-Maximization algorithm is proposed to run the model’s inference as well as a model selection criterion to choose the number of coclusters. Both simulated and real data sets illustrate the eciency of this model by its ability to easily identify relevant co-clusters

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Factorizations of complete graphs into brooms

Let r and n be positive integers with r<2n. A broom of order 2n is the union of the path on P2n−r−1 and the star K1,r, plus one edge joining the center of the star to an endpoint of the path. It was shown by Kubesa (2005) [10] that the broom factorizes the complete graph K2n for odd n and View the MathML source. In this note we give a complete classification of brooms that factorize K2n by giving a constructive proof for all View the MathML source (with one exceptional case) and by showing that the brooms for View the MathML source do not factorize the complete graph K2n.Web of Science31261093108