We consider the problem of labeling the nodes of an n-node graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and AT-free graphs). We also show lower bounds, and prove that most of our schemes are optimal in length of labels generated and in the quality of the approximation, leaving some open problems.

In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.

We show that there is an O(n^3) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n log n) algorithm to approximate the bandwidth of an AT-free graph within a factor 4 and an O(n+ e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid graphs we obtain O(n log² n) algorithms with worst case performance ratio 2. For cocomparability graphs we obtain an O(n + e) algorithm with worst case performance ratio 3. Finally, we show that there is an O(n² log² n) algorithm to compute the exact bandwidth of chain graphs.

An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that AT-free graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected AT-free graphs. The resulting simple algorithm, based on the well-known Lexicographic Breadth-First Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).

Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusion-minimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algorithm which efficiently computes a minimal triangulation using an arbitrary ordering on the vertices. 1 Introduction. Computing a minimal triangulation consists in embedding a given graph into a triangulated graph by adding a set of edges (called a fill). If the set of edges added is inclusion-minimal, the fill is said to be minimal, and the corresponding triangulated graph is called a minimal triangulation. Finding a fill that is minimum is NP-complete ([10]). Given a graph G and any ordering ff on its vertices, an associated fill can be computed by repeatedly choosing the next vertex x in order ff, adding the edges necessary to make the neighborhood of x into a clique (i.e. by making x si...

A spanning tree of a graph is a k-additive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distance-hereditary graphs, interval graphs, asteroidal-triple free graphs, allow some constant k such that every member of the class has some k-additive tree spanner. On the other hand, there are chordal graphs without k-additive tree spanner for arbitrary large k.

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithms to solve the<F3.728e+05> independent dominating set<F3.817e+05> and the<F3.728e+05> independent perfect dominating set<F3.817e+05> problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems clique and partition into cliques remain NP-complete when restricted to AT-free graphs.

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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSP-graph (there exists a dominating shortest path) admits an additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.