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77
Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 87 (13 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
LexBFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing
, 2000
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Asteroidal TripleFree Graphs
, 1997
"... . An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in ..."
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Cited by 54 (10 self)
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. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triplefree graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of ATfree graphs. Specifically, we show that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of ATfree graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for ATfree graphs. An assortment of other properties of ATfree graphs is also p...
Graph Sandwich Problems
, 1994
"... The graph sandwich problem for property \Pi is defined as follows: Given two graphs G ) such that E ` E , is there a graph G = (V; E) such that E which satisfies property \Pi? Such problems generalize recognition problems and arise in various applications. Concentrating mainly o ..."
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Cited by 48 (8 self)
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The graph sandwich problem for property \Pi is defined as follows: Given two graphs G ) such that E ` E , is there a graph G = (V; E) such that E which satisfies property \Pi? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NPcompleteness of others. We describe
LinearTime Recognition of CircularArc Graphs
 Algorithmica
, 2003
"... A graph G is a circulararc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a lineartime algorithm for recognizing this class of graphs. W ..."
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Cited by 36 (7 self)
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A graph G is a circulararc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a lineartime algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.
Approximating the Bandwidth for Asteroidal TripleFree Graphs
"... We show that there is an O(n^3) algorithm to approximate the bandwidth of an ATfree 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 ATfree graph within a factor 4 an ..."
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Cited by 35 (1 self)
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We show that there is an O(n^3) algorithm to approximate the bandwidth of an ATfree 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 ATfree 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.
PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 34 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Optimal FPGA Module Placement with Temporal Precedence Constraints
 IN PROC. DATE 2001, DESIGN, AUTOMATION AND TEST IN EUROPE
, 2001
"... We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as threedimensional boxes in space and time. Using a graphtheoretic characterization of feasible packings, we are able to solve the following p ..."
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Cited by 33 (5 self)
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We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as threedimensional boxes in space and time. Using a graphtheoretic characterization of feasible packings, we are able to solve the following problems: (a) Find the minimal execution time of the given problem on an FPGA of fixed size, (b) Find the FPGA of minimal size to accomplish the tasks within a fixed time limit. Furthermore, our approach is perfectly suited for the treatment of precedence constraints for the sequence of tasks, which are present in virtually all practical instances. Additional mathematical structures are developed that lead to a powerful framework for computing optimal solutions. The usefulness is illustrated by computational results.
Certifying algorithms for recognizing interval graphs and permutation graphs
 SIAM J. COMPUT
, 2006
"... A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition o ..."
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Cited by 31 (7 self)
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A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(V) time.
List Homomorphisms and Circular Arc Graphs
 Combinatorica
, 1999
"... List homomorphisms generalize list colourings in the following way: Given graphs G; H , and lists L(v) ` V (H); v 2 V (G), a list homomorphism of G to H with respect to the lists L is a mapping f : V (G) ! V (H) such that uv 2 E(G) implies f(u)f(v) 2 E(H), and f(v) 2 L(v) for all v 2 V (G). The l ..."
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Cited by 30 (6 self)
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List homomorphisms generalize list colourings in the following way: Given graphs G; H , and lists L(v) ` V (H); v 2 V (G), a list homomorphism of G to H with respect to the lists L is a mapping f : V (G) ! V (H) such that uv 2 E(G) implies f(u)f(v) 2 E(H), and f(v) 2 L(v) for all v 2 V (G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists L(v) ` V (H), v 2 V (G), admits a list homomorphism with respect to L. The list homomorphism problem was introduced by Feder and Hell, who proved that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NPcomplete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NPcomplete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Key Words: Homomorphisms, listhomomorphisms, retractions, asteroidal triples, circular arc graphs, algorithms, complexity. 1