The complexity of two graph orientation problems

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol

Minimizing the oriented diameter of a planar graph

We consider the problem of minimizing the diameter of an orientation of a planar graph. A result of Chvátal and Thomassen shows that for general graphs, it is NP-complete to decide whether a graph can be oriented so that its diameter is at most two. In contrast to this, for each constant l, we describe an algorithm that decides if a planar graph G has an orientation with diameter at most l and runs in time O(c|V|), where c depends on l

The clustering coefficient of a scale-free random graph

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to (log n)/n. Bollobas and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ((log n)^2)/n

Improved bounds for the number of forests and acyclic orientations in the square lattice

In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice $L_n$. The authors gave the following bounds for the asymptotics of $f(n)$, the number of forests of $L_n$, and $\alpha(n)$, the number of acyclic orientations of $L_n$: $3.209912 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.84161$ and $22/7 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.70925$. In this paper we improve these bounds as follows: $3.64497 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.74101$ and $3.41358 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.55449$. We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices

Review of \u3cem\u3eKinship with Strangers: Adoption and Interpretations of Kinship in American Culture.\u3c/em\u3e Judith S. Modell. Reviewed by Dorina N. Novle, Louisiana State University.

Judith S. Modell. Kinship with Strangers: Adoption and Interpretations of Kinship in American Culture. Berkeley, CA: University of California Press, 1994. $35 hardcover

Review of \u3cem\u3eA Sealed and Sacred Kinship: The Culture of Policies and Practices in American Adoption.\u3c/em\u3e Judith S. Modell. Reviewed by Dorinda N. Noble.

Book review of Judith S. Modell, A Sealed and Sacred Kinship: The Culture of Policies and Practices in American Adoption. New York: Berghahn, 2002. $49.95 hardcover

Review of \u3cem\u3eChild Protection Systems: International Trends and Orientations.\u3c/em\u3e Neil Gilbert, Nigel Parton, & Marit Skivenes, (Eds.). Reviewed by Dorinda N. Noble.

Book review of Neil Gilbert, Nigel Parton, & Marit Skivenes, Eds. (2011). Child Protection Systems: International Trends and Orientations. New York: Oxford University Press, $55.00 (hardcover)