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12
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 55 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show th ..."
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Cited by 29 (6 self)
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We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NPhard, and W [t]hard for all t. We also show that the second problem is W [1]hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.
Computing Optimal Linear Layouts of Trees in Linear Time
 Proc. ESA 2000, number 1879
, 1999
"... We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the ru ..."
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Cited by 14 (0 self)
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We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].
Finiteness Theorems for Graphs and Posets Obtained by Compositions
 Order
, 1998
"... We investigate classes of graphs and posets that admit decompositions to obtain or disprove finiteness results for obstruction sets. To do so we develop a theory of minimal infinite antichains that allows us to characterize such antichains by means of the set of elements below it. In particular we ..."
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Cited by 4 (0 self)
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We investigate classes of graphs and posets that admit decompositions to obtain or disprove finiteness results for obstruction sets. To do so we develop a theory of minimal infinite antichains that allows us to characterize such antichains by means of the set of elements below it. In particular we show that the following classes have infinite antichains with resp. to the induced subgraph/poset relation: interval graphs and orders, Nfree orders, orders with bounded decomposition width. On the other hand for orders with bounded decomposition diameter finiteness of all antichains is shown. As a consequence those classes with infinite antichains have undecidable hereditary properties whereas those with finite antichains have fast algorithms for all such properties.
Graph Searching on Chordal Graphs
, 1997
"... In the graph searching problem, initially a graph with all edges contaminated is presented. We would like to obtain a state of the graph in which all edges are simultaneously clear by a sequence of moves using searchers. The objective is to achieve the desired state by using the least number of sear ..."
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Cited by 3 (0 self)
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In the graph searching problem, initially a graph with all edges contaminated is presented. We would like to obtain a state of the graph in which all edges are simultaneously clear by a sequence of moves using searchers. The objective is to achieve the desired state by using the least number of searchers. Two variations of the graph searching problem are considered in this paper. One is the edge searching, in which the clearing of an edge is accomplished by moving a searcher along the edge, and the other is the node searching, in which an edge is cleared by concurrently having searchers on both of its endpoints. In this paper, we present a uniform approach to solve the above two graph searching problems on several classes of chordal graphs. For edge searching problem, we give an O(mn 2 )time algorithm on split graphs, an O(m + n)time algorithm on interval graphs, and an O(mn k )time algorithm on kstarlike graphs (a generalization of split graphs), for a fixed k 2, where m and ...
Mixed search number and linearwidth of interval and split graphs
 Proceedings of WG 2007, Lecture Notes in Computer Science 4769, 2007
"... Mixed search number and linearwidth of interval and split graphs ..."
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Cited by 3 (1 self)
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Mixed search number and linearwidth of interval and split graphs
Treewidth and Small Separators for Graphs with Small Chordality
, 1995
"... A graph G kchordal, if it does not contain chordless cycles of length larger than k. The chordality cl of a graph G is the minimum k for which G is kchordal. The degeneracy or the width of a graph is the maximum mindegree of any of its subgraphs. Our results are the following: 1. The problem of ..."
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Cited by 3 (1 self)
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A graph G kchordal, if it does not contain chordless cycles of length larger than k. The chordality cl of a graph G is the minimum k for which G is kchordal. The degeneracy or the width of a graph is the maximum mindegree of any of its subgraphs. Our results are the following: 1. The problem of treewidth remains NPcomplete when restricted on graphs with small maximum degree. 2. An upper bound is given for the treewidth of a graph as a function of its maximum degree and chordality. A consequence of this result is that when maximum degree and chordality are fixed constants, then there is a linear algorithm for treewidth and a polynomial algorithm for pathwidth. 3. For any constant s 1, it is shown that any (s + 2)chordal graph with degeneracy d contains a 1 2 separator of size O((dn) s\Gamma1 s ), computable in linear time. Our results extent the many applications of the separator theorems in [1, 33, 34] to the class of kchordal graphs. Several natural classes of graphs have ...
Some Results on Nondeterministic Graph Searching in Trees ∗
"... Pathwidth and treewidth of graphs have been extensively studied for their important structural and algorithmic aspects. Determining these parameters is NPcomplete in general, however it becomes linear time solvable when restricted to some special classes of graphs. In particular, many algorithms ha ..."
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Cited by 1 (0 self)
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Pathwidth and treewidth of graphs have been extensively studied for their important structural and algorithmic aspects. Determining these parameters is NPcomplete in general, however it becomes linear time solvable when restricted to some special classes of graphs. In particular, many algorithms have been proposed to compute efficiently the pathwidth of trees. Skodinis (2000) proposes a linear time algorithm for this task. Pathwidth and treewidth have also been studied for their nice gametheoretical interpretation, namely graph searching games. Roughly speaking, graph searching problems look for the smallest number of searchers that are sufficient to capture a fugitive in a graph. Fomin et al. (2005) define the nondeterministic graph searching that provides an unified approach for the pathwidth and the treewidth of a graph. Given q ≥ 0, the qlimited search number, denotes by sq(G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, such that the searchers are allowed to know the position of the fugitive at most q times. Roughly, s0(G) corresponds to the pathwidth of a graph G, and s∞(G) corresponds to its treewidth. Fomin et al. proved that computing sq(G) is NPcomplete in general, and left open the complexity of the problem restricted to the class of trees. This paper studies this latter problem. On one hand, we give tight upper bounds on the number of queries required to search a tree when the number of searchers is fixed. We also prove that this number can be computed in linear time when two searchers are used. On the other hand, our main result consists in the design of a simple polynomial time algorithm that computes a 2approximation of sq(T), for any tree T and any q ≥ 0. This algorithm becomes exact if q ∈ {0, 1}, which proves that the decision problem associated to s1 is polynomial in the class of trees. 1
Edge Clique Graphs and Some Classes of Chordal Graphs
, 2002
"... The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlikethreshold, s ..."
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The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlikethreshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal cliques ought to be triangular numbers. We show that this condition is also su7cient for starlikethreshold graphs.