Given an undirected multigraph G = (V, E), a family W of sets W ⊆ V of vertices (areas), and a requirement function r : W → Z + (where Z + is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area W ∈ W. So far this problem was shown to be NP-hard in the uniform case of r(W) = 1 for each W ∈ W, and polynomially solvable in the uniform case of r(W) = r ≥ 2 for each W ∈ W. In this paper, we show that the problem can be solved in O(m+ pn 4 (r * + log n)) time, even if r(W) ≥ 2 holds for each W ∈ W, where n = |V |, m = |{{u, v}|(u, v) ∈ E}|, p = |W|, and r * = max{r(W) | W ∈ W}.

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An L(2, 1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that | f (x) − f (y)| ≥ 2 if x and y are adjacent and | f (x) − f (y)| ≥ 1 if x and y are at distance 2 for all x and y in V(G). A k-L(2, 1)labeling is an assignment f : V(G) → {0, . . . , k}, and the L(2, 1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an O(∆ 4.5 n) time algorithm for a tree T has been known so far, where ∆ is the maximum degree of T and n = |V(T )|. In this paper, we first show that an existent necessary condition for λ(T ) = ∆ + 1 is also sufficient for a tree T with ∆ = Ω( √ n), which leads a linear time algorithm for computing λ(T ) under this condition. We then show that λ(T ) can be computed in O(∆ 1.5 n) time for any tree T . Combining these, we finally obtain an O(n 1.75 ) time algorithm, which greatly improves the currently best known result.

Given a graph G = (V, E) and an integer D ≥ 1, we consider the problem of augmenting G by the smallest number of new edges so that the diameter becomes at most D. It is known that no constant approximation algorithms to this problem with an arbitrary graph G can be obtained unless P = N P . For a forest G and an odd D ≥ 3, it was open whether the problem is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with a forest G and an odd D; our algorithm delivers an 8-approximate solution in O(|V | 3 ) time. We also show that a 4-approximate solution to the problem with a forest G and an odd D can be obtained in linear time if the augmented graph is additionally required to be biconnected.

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