Babak Farzad

Complexity of Finding Graph Roots with Girth Conditions

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G=H2 for some graph H of small girth. The main results are the following. There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism.There is a polynomial time algorithm to recognize if G=H2 for some graph H of girth at least 6.It is NP-complete to recognize if G=H2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

COMPUTING GRAPH ROOTS WITHOUT SHORT CYCLES

Gra ph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H 2 , however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H 2 for some graph H of small girth. The main results are These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

On the edge-density of 4-critical graphs

Gallai conjectured that every 4-critical graph on n vertices has at least 5/3n-2/3 edges. We prove this conjecture for 4-critical graphs in which the subgraph induced by vertices of degree 3 is connected.

Dynamics of large scale networks following a merger

We studied the dynamic network of relationships among avatars in the massively multiplayer online game Planetside 2. In the spring of 2014, two separate servers of this game were merged, and as a result, two previously distinct networks were combined into one. We observed the evolution of this network in the seven month period following the merger. We found that some structures of original networks persist in the combined network for a long time after the merger. As the original avatars are gradually removed, these structures slowly dissolve, but they remain observable for a surprisingly long time. We present a number of visualizations illustrating the post-merger dynamics and discuss time evolution of selected quantities characterizing the topology of the network.

Backbone colourings of graphs

Consider an undirected graph G and a subgraph of G , called H . A backbone k -colouring of ( G , H ) is a colouring f : V ( G ) ź { 1 , 2 , ź , k } such that G is properly coloured and for each edge of H , the colours of its endpoints differ by at least 2. The minimum number k for which there is a backbone k -colouring of ( G , H ) is the backbone chromatic number BBC ( G , H ) .We prove that every graph with chromatic number k has a proper k -colouring and a spanning tree T such that the colours on the endpoints of edges of T differ by exactly 1. As a corollary of this result, we show that for any graph G with ź ( G ) ź 4 there exists a spanning tree T of G such that BBC ( G , T ) = ź ( G ) .

Three-colourability of planar graphs with no 5- or triangular {3,6}-cycles

Abstract Steinbergs conjecture asserts that every planar graph without 4- and 5-cycles is 3-colourable. In this paper, we prove that planar graphs without 5-cycles and without triangles adjacent to 3- and 6-cycles are 3-colourable.

Social and Economic Network Formation: A Dynamic Model

We study the dynamics of a game-theoretic network formation model that yields large-scale small-world networks. So far, mostly stochastic frameworks have been utilized to explain the emergence of these networks. On the other hand, it is natural to seek for game-theoretic network formation models in which links are formed due to strategic behaviors of individuals, rather than based on probabilities. Inspired by Even-Dar and Kearns’ model [8], we consider a more realistic framework in which the cost of establishing each link is dynamically determined during the course of the game. Moreover, players are allowed to put transfer payments on the formation and maintenance of links. Also, they must pay a maintenance cost to sustain their direct links during the game. We show that there is a small diameter of at most 4 in the general set of equilibrium networks in our model. We achieved an economic mechanism and its dynamic process for individuals which firstly; unlike the earlier model, the outcomes of players’ interactions or the equilibrium networks are guaranteed to exist. Furthermore, these networks coincide with the outcome of pairwise Nash equilibrium in network formation. Secondly; it generates large-scale networks that have a rational and strategic microfoundation and demonstrate the main characterization of small degree of separation in real-life social networks. Furthermore, we provide a network formation simulation that generates small-world networks.

Random Lifts of

Amit, Linial, and Matousek [Random Struct. Algorithms, 20 (2001), pp. 1-22] have raised the following question: Is the chromatic number of random

K_5\backslashe

h

are 3-Colorable

-lifts of