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89
Qualitative Spatial Reasoning: Cardinal Directions as an Example
, 1996
"... Geographers use spatial reasoning extensively in largescale spaces, i.e., spaces that cannot be seen or understood from a single point of view. Spatial reasoning differentiates several spatial relations, e.g. topological or metric relations, and is typically formalized using a Cartesian coordinate ..."
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Cited by 116 (7 self)
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Geographers use spatial reasoning extensively in largescale spaces, i.e., spaces that cannot be seen or understood from a single point of view. Spatial reasoning differentiates several spatial relations, e.g. topological or metric relations, and is typically formalized using a Cartesian coordinate system and vector algebra. This quantitative processing of information is clearly different from the ways humans draw conclusions about spatial relations. Formalized qualitative reasoning processes are shown to be a necessary part of Spatial Expert Systems and Geographic Information Systems. Addressing a subset of the total problem, namely reasoning with cardinal directions, a completely qualitative method, without recourse to analytical procedures, is introduced and a method for its formal comparison with quantitative formulae is defined. The focus is on the analysis of cardinal directions and their properties. An algebraic method is used to formalize the meaning of directions. The standard...
Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Cited by 100 (2 self)
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
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"... A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (M ..."
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Cited by 71 (4 self)
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A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the socalled Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multisweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2sweep MNS certifying recognition algorithm. Submitted:
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 55 (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
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k ..."
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Cited by 49 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
Simple Linear Time Recognition of Unit Interval Graphs
, 1998
"... We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whe ..."
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Cited by 34 (1 self)
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We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whenever G is a unit interval graph. This order corresponds to the order of the intervals of some unit interval model for G when arranged according to the increasing order of their left end coordinates. BreadthFirst Search can also be used to construct a unit interval model for a unit interval graph on n vertices; in this model each endpoint is rational, with denominator n. Keywords: graph algorithms, interval graphs, BreadthFirst Search, unit interval graphs, proper interval graphs, design of algorithms. 1 Introduction A graph G is an interval graph if its vertices can be put in a one to one correspondence with a family of intervals I on the real line such that two vertices in G are a...
Nested Graphs: A Graphbased Knowledge Representation Model with FOL Semantics
 Proceedings of the 6th International Conference on Knowledge Representation (KR'98
, 1998
"... We present a graphbased KR model issued from Sowa's conceptual graphs but studied and developed with a speci c approach. Formal objects are kinds of labelled graphs, which maybesimple graphs or nested graphs. The fundamental notion for doing reasonings, called projection (or subsumption), is a ..."
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Cited by 33 (9 self)
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We present a graphbased KR model issued from Sowa's conceptual graphs but studied and developed with a speci c approach. Formal objects are kinds of labelled graphs, which maybesimple graphs or nested graphs. The fundamental notion for doing reasonings, called projection (or subsumption), is a kind of labelled graph morphism. Thus, we propose a graphical KR model, where \graphical &quot; is used in the sense of [Sch91], i.e. a model that \uses graphtheoretic notions in an essential and nontrivial way&quot;. Indeed, morphism, which is the fundamental notion for any structure, is at the core of our theory. We de ne two rst order logic semantics, which correspond to di erentintuitivesemantics, and proveinboth cases that projection is sound and complete with respect to deduction. This paper is almost identical to the paper appeared in the KR'98 proceedings. It provides minor corrections. 1
A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs
 SIAM J. COMPUT
, 1999
"... In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to ..."
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Cited by 27 (1 self)
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In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusionfree family of intervals. This problem has important applications in physical mapping of DNA. We give a nearoptimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with wordsize b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.
Representing firstorder logic using graphs
 International Conference on Graph Transformations (ICGT), volume 3256 of Lecture Notes in Computer Science
, 2004
"... Abstract. We show how edgelabelled graphs can be used to represent firstorder logic formulae. This gives rise to recursively nested structures, in which each level of nesting corresponds to the negation of a set of existentials. The model is a direct generalisation of the negative application cond ..."
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Cited by 20 (8 self)
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Abstract. We show how edgelabelled graphs can be used to represent firstorder logic formulae. This gives rise to recursively nested structures, in which each level of nesting corresponds to the negation of a set of existentials. The model is a direct generalisation of the negative application conditions used in graph rewriting, which count a single level of nesting and are thereby shown to correspond to the fragment ∃¬ ∃ of firstorder logic. Vice versa, this generalisation may be used to strengthen the notion of application conditions. We then proceed to show how these nested models may be flattened to (sets of) plain graphs, by allowing some structure on the labels. The resulting formulaeasgraphs may form the basis of a unification of the theories of graph transformation and predicate transformation. 1
Certifying LexBFS recognition algorithms for proper inteval graphs and proper interval bigraphs
 SIAM J. Discrete Math
"... Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the r ..."
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Cited by 16 (3 self)
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Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m + n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n), the certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). 1