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We consider the problem of coding labelled trees by means of strings of vertex labels and we present a general scheme to define bijective codes based on the transformation of a tree into a functional digraph. Looking at the fields in which codes for labelled trees are utilized, we see that the properties of locality and heritability are required and that codes like the well known Prüfer code do not satisfy these properties. We present a general scheme for generating codes based on the construction of functional digraphs. We prove that using this scheme, locality and heritability are satisfied as a direct function of the similarity between the topology of the functional digraph and that of the original tree. Moreover, we also show that the efficiency of our method depends on the transformation of the tree into a functional digraph. Finally we show how it is possible to fit three known codes into our scheme, obtaining maximum efficiency and high locality and heritability

A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers dmin and dmax such that each leaf lu of T corresponds to a vertex u ∈ V and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (lu, lv) ≤ dmax where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T . In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where dmin = 0 (LPG) and dmax = +∞ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.

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A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge weight function w on T, and two non-negative real numbers d(mm) <= d(max), such that each leaf l(u) of T corresponds to a vertex u is an element of V and there is an edge (u, V) is an element of E if and only if d(min) <= d(T,w)(l(u), l(v)) <= d(max) where d(T,w)(l(u), l(v)) is the sum of the weights of the edges on the unique path from l(u), to l(v) in T. In this paper we analyze the class of PCGs in relation to two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves concern only d(max) (LPG) or only d(min) (mLPG). In particular, we show that the union of LPG and mLPG classes does not coincide with the whole class of PCGs, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, we study the closure properties of the classes PCG, mLPG and LPG, under some common graph operations. In particular, we consider the following operations: adding an isolated or universal vertex, adding a pendant vertex, adding a false or a true twin, taking the complement of a graph and taking the disjoint union of two graphs. (C) 2012 Elsevier B.V. All rights reserved

Abstract. Topology recognition is one of the fundamental distributed tasks in networks. Each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ≤ αn, for any constant α < 1. In most cases, our bounds are asymptotically tight. More precisely, if the allotted time is D − k, where 0 < k ≤ D, then the optimal size of advice is Θ((n 2 log n)/(D − k + 1)). If the allotted time is D, then this optimal size is Θ(n log n). If the allotted time is D + k, where 0 < k ≤ D/2, then the optimal size of advice is Θ(1 + (log n)/k). The only remaining gap between our bounds is for time D + k, where D/2 < k ≤ D. In this time interval our upper bound remains O(1 + (log n)/k), while the lower bound (that holds for any time) is 1. This leaves a gap if D ∈ o(log n). Finally, we show that for time 2D + 1, one bit of advice is both necessary and sufficient. Our results show how sensitive is the minimum size of advice to the time allowed for topology recognition: allowing just one round more, from D to D + 1, decreases exponentially the advice needed to accomplish this task.

Among all simple graphs on n vertices and e edges, which ones have the largest sum of squares of the vertex degrees? It is easy to see that they must be threshold graphs, but not every threshold graph is optimal in this sense. Boesch et al. [Boesch et al., Tech Rep, Stevens Inst Tech, Hoboken NJ, 1990] showed that for given n and e there exists exactly one graph of the form G 1 (p, q, r) = K p + (S q ∪ K 1,r ) and exactly one G 2 (p, q, r) = S p ∪ (K q + K 1,r ) and that one of them is optimal, where K and S indicate complete and edgeless graphs, K 1,r indicates a star on r + 1 vertices, ∪ indicates disjoint union, and + indicates complete disjoint join. We specify a general threshold graph in the form G *or its complement * Part of this research was conducted while the first author was visiting the University of Rome "La Sapienza" and theÉcole Polytechnique Fédérale de Lausanne, whose support is gratefully acknowledged.

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