scholarly journals A four-sweep LBFS recognition algorithm for interval graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Structure Theory ◽  
Time Interval ◽  
Breadth First Search ◽  
Input Graph ◽  
Graph Recognition ◽  
Unit Interval Graph ◽  
International Audience

Graph Theory International audience In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

scholarly journals Recognizing HH-free, HHD-free, and Welsh-Powell Opposition Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Leonidas Palios
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Recognition Algorithm ◽  
Space Complexity ◽  
Linear Time Algorithm ◽  
Free Graph ◽  
Graph Recognition ◽  
International Audience ◽  
Time And Space Complexity ◽  
Perfect Elimination Ordering

International audience In this paper, we consider the recognition problem on three classes of perfectly orderable graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min \m α (n,n), m + n^2 log n\) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min \m α (n,n), m + n^2 log n\)-time and O(n+m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

2000 ◽  
Vol 11 (03) ◽  
pp. 423-443 ◽  
Author(s):  
MARTIN CHARLES GOLUMBIC ◽  
UDI ROTICS
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Unit Interval ◽  
Perfect Graphs ◽  
The Other ◽  
Perfect Graph ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Unit Interval Graph

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

A linear-time algorithm for proper interval graph recognition

1995 ◽  
Vol 56 (3) ◽  
pp. 179-184 ◽  
Author(s):  
Celina M.Herrera de Figueiredo ◽  
João Meidanis ◽  
Célia Picinin de Mello
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Interval Graph ◽  
Linear Time Algorithm ◽  
Graph Recognition ◽  
Proper Interval Graph

scholarly journals Closed graphs are proper interval graphs

2014 ◽  
Vol 22 (3) ◽  
pp. 37-44
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Simple Graph ◽  
Interval Graphs ◽  
Closed Graph ◽  
Graph Recognition ◽  
Proper Interval Graph ◽  
Linear Time Algorithms

Abstract Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.

scholarly journals A new characterization and a recognition algorithm of Lucas cubes

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Andrej Taranenko
Keyword(s):  
Graph Theory ◽  
Interconnection Networks ◽  
Linear Time ◽  
Recognition Algorithm ◽  
Induced Subgraphs ◽  
Fibonacci Cube ◽  
Vertex Set ◽  
International Audience ◽  
Binary Strings

Graph Theory International audience Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.

scholarly journals Linear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance Graphs and Bipartite Probe Interval Graphs

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
David E. Brown ◽  
Arthur H. Busch ◽  
Garth Isaak
Keyword(s):  
Real Line ◽  
Positive Real Number ◽  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Positive Real ◽  
Recognition Algorithms ◽  
Probe Interval Graphs ◽  
Tolerance Graphs ◽  
International Audience

Graphs and Algorithms International audience A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.

scholarly journals Recognition and Characterization of Chronological Interval Digraphs

10.37236/2497 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Sandip Das ◽  
Mathew Francis ◽  
Pavol Hell ◽  
Jing Huang
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Interval Graphs ◽  
Structure Characterization ◽  
Left Endpoint ◽  
New Class ◽  
Closed Intervals ◽  
Novel Structure ◽  
Definition Of

Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time recognition algorithm. In this paper we identify another natural digraph analogue of interval graphs that we call ”chronological interval digraphs”. By contrast, the new class admits both a forbidden structure characterization and a linear time recognition algorithm. Chronological interval digraphs arise by interpreting the standard definition of an interval graph with a natural orientation of its edges. Specifically, $G$ is a chronological interval digraph if there exists a family of closed intervals $I_v$, $v \in V(G)$, such that $uv$ is an arc of $G$ if and only if $I_u$ intersects $I_v$ and the left endpoint of $I_u$ is not greater than the left endpoint of $I_v$. (Equivalently, if and only if $I_u$ contains the left endpoint of $I_v$.)We characterize chronological interval digraphs in terms of vertex orderings, in terms of forbidden substructures, and in terms of a novel structure of so-called $Q$-paths. The first two characterizations exhibit strong similarity with the corresponding characterizations of interval graphs. The last characterization leads to a linear time recognition algorithm.

scholarly journals Finding hidden cliques in linear time

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Uriel Feige ◽  
Dorit Ron
Keyword(s):  
High Probability ◽  
Linear Time ◽  
Maximum Clique ◽  
Spectral Approach ◽  
Input Graph ◽  
Random Choice ◽  
Spectral Techniques ◽  
Vertex Graph ◽  
International Audience ◽  
Linear Time Algorithms

International audience In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.

AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

2000 ◽  
Vol 11 (03) ◽  
pp. 365-371 ◽  
Author(s):  
LJUBOMIR PERKOVIĆ ◽  
BRUCE REED
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Tree Decomposition ◽  
Linear Time Algorithm ◽  
Input Graph ◽  
Primary Motivation ◽  
Small Width ◽  
Tree Decompositions ◽  
Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

Multivariate linear time series models

1984 ◽  
Vol 16 (3) ◽  
pp. 492-561 ◽  
Author(s):  
E. J. Hannan ◽  
L. Kavalieris
Keyword(s):  
Limit Theorems ◽  
Analysis Of Algorithms ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Real Data ◽  
Rates Of Convergence ◽  
Structure Theory ◽  
Time Series Models ◽  
Transfer Model ◽  
Rational Transfer

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

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