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 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 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 = H 2 for some graph H of girth at least 6.• It is NP-complete to recognize if G = H 2 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.

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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 (NIPS 19: 385-392, 2007), 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 four 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. Moreover, we provide a network formation simulation that generates small-world networks.

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

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 H 2 . Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan [11] proved that it is NP-complete to determine if a given graph G is the square of some graph. The graph introduced in their reduction is a graph that contains many triangles and is relatively dense. Farzad et al. [5] proved the NP-completeness for finding a square root for girth 4 while they gave a polynomial time algorithm for computing a square root of girth at least six. Adamaszek and Adamaszek [1] proved that if a graph has a square root of girth six then this square root is unique up to isomorphism. In this paper we consider the characterization and recognition problem of graphs that are square of graphs of girth at least five. We introduce a family of graphs with exponentially many non-isomorphic square roots, and as the main result of this paper we prove that the square root finding problem is NP-complete for square roots of girth five. This proof is providing the complete dichotomy theorem for square root problem in terms of the girth of the square roots.