Aleksandar Ilic

Finding critical regions and region-disjoint paths in a network

Due to their importance to society, communication networks should be built and operated to withstand failures. However, cost considerations make network providers less inclined to take robustness measures against failures that are unlikely to manifest, like several failures coinciding simultaneously in different geographic regions of their network. Considering networks embedded in a two-dimensional plane, we study the problem of finding a critical region-a part of the network that can be enclosed by a given elementary figure of predetermined size-whose destruction would lead to the highest network disruption. We determine that only a polynomial, in the input, number of nontrivial positions for such a figure needs to be considered and propose a corresponding polynomial-time algorithm. In addition, we consider region-aware network augmentation to decrease the impact of a regional failure. We subsequently address the region-disjoint paths problem, which asks for two paths with minimum total weight between a source (s) and a destination (d) that cannot both be cut by a single regional failure of diameter D (unless that failure includes s or d). We prove that deciding whether region-disjoint paths exist is NP-hard and propose a heuristic region-disjoint paths algorithm.

On the distance based graph entropies

In this note we prove a sharp upper bound for the distance based graph entropy introduced in Chen et?al. (2014). Furthermore we resolve one part of the conjecture proposed in the same paper and provide counterexamples for the other part.

On the extremal values of general degree-based graph entropies

In this note we prove one part of the conjecture about upper and lower bounds of the degree-based graph entropy Ik(T) in the class of trees introduced in S. Cao, M. Dehmer, Y. Shi, Extremality of degree-based graph entropies, Information Sciences 278 (2014) 22-33. using Lagrange multipliers and Jensens inequality, and disprove the other part by providing a family of counter-examples. Our main result is the following: the path Pn is unique tree on n vertices that maximizes Ik(T) for k > 0, and the star Sn is unique tree on n vertices that minimizes Ik(T) for k ? 1.

On the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider a special case of the ordinary NP-hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. In Panwalkar and Koulamas (2012) 5], the authors presented quadratic algorithm for the problem when each job has its smaller processing time on the first machine. In this note, we improve the running time of the algorithm to O ( n log ? n ) by efficient implementation using recently introduced modified binary tree data structure.

Bipartite graphs with bounded vertex degree and largest eigenvalue of the form r

In this note, we generalize two results from [2] and [3] and for every Δ and Δ ≤ r ≤ Δ2 construct a family of bipartite graphs with the maximal vertex degree Δ and largest eigenvalue equal to r.

Counterexamples to conjectures on graph distance measures based on topological indexes

In this paper we disprove three conjectures from [M. Dehmer, F. Emmert-Streib, Y. Shi, Interrelations of graph distance measures based on topological indices, PLoS ONE 9 (2014) e94985] on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities. The constructions are based on the families of trees that have the same Wiener index, graph energy or Randic index - but different degree sequences.