For every d ≥ 3 and k ∈ {2} ∪ [3, ∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular colorings.

a b s t r a c tAn induced matching in a graph G = (V , E) is a matching M such that (V , M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al.[10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n 27 + O(1). We improve both these bounds to n 28 + O(1), which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity. Kanj et al. also presented an algorithm which decides in O(2 159 √ k + n)-time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient O(2 25.5 √ k + n)-time algorithm. Its main ingredient is a new, O * (4 l )-time algorithm for finding a maximum induced matching in a graph of branch width at most l.

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Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show N P-completeness. We complement our results by providing linear time algorithms for these searches on split graphs. * The work of this paper was done in the framework of a bilateral project between Brandenburg

The eccentricity of a node v in a network is the maximum distance from v to any other node. In social networks, the reciprocal of eccentricity is used as a measure of the importance of a node within a network. The associated centralization measure then calculates the degree to which a network is dominated by a particular node. In this work, we determine the maximum value of eccentricity centralization as well as the most centralized networks for various classes of networks including the families of bipartite networks (twomode data) with given partition sizes and tree networks with fixed number of nodes and fixed maximum degree. To this end, we introduce and study a new way of enumerating the nodes of a tree which might be of independent interest.