scholarly journals Recognizing Circulant Graphs in Polynomial Time: An Application of Association Schemes

10.37236/1570 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Polynomial Time ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Circulant Graphs ◽  
Symmetry Properties ◽  
Coherent Configurations

In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism.

scholarly journals Recognizing Circulant Graphs of Prime Order in Polynomial Time

10.37236/1363 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Adjacency Matrix ◽  
Prime Order ◽  
Circulant Matrix ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Necessary Condition ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Coherent Configuration ◽  
Schur Ring

A circulant graph $G$ of order $n$ is a Cayley graph over the cyclic group ${\bf Z}_n.$ Equivalently, $G$ is circulant iff its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration ${\cal A}$ and, in particular, a Schur ring ${\cal S}$ isomorphic to ${\cal A}$. ${\cal A}$ can be associated without knowing $G$ to be circulant. If $n$ is prime, then by investigating the structure of ${\cal A}$ either we are able to find an appropriate ordering of the vertices proving that $G$ is circulant or we are able to prove that a certain necessary condition for $G$ being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in $n$.

scholarly journals Indirect identification of horizontal gene transfer

2021 ◽  
Vol 83 (1) ◽  
Author(s):  
David Schaller ◽  
Manuel Lafond ◽  
Peter F. Stadler ◽  
Nicolas Wieseke ◽  
Marc Hellmuth
Keyword(s):  
Gene Transfer ◽  
Horizontal Gene Transfer ◽  
Polynomial Time ◽  
Divergence Time ◽  
Recognition Algorithm ◽  
Vertex Coloring ◽  
Implicit Methods ◽  
Colored Graphs ◽  
Wide Range ◽  
Complete Multipartite

AbstractSeveral implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

scholarly journals A Quasi-Hole Detection Algorithm for Recognizing k-Distance-Hereditary Graphs, with k < 2

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 105
Author(s):  
Serafino Cicerone
Keyword(s):  
Polynomial Time ◽  
Detection Algorithm ◽  
Recognition Algorithm ◽  
Forbidden Subgraphs ◽  
Induced Subgraph ◽  
Hole Detection

Cicerone and Di Stefano defined and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the k-distance-hereditary graphs such that k<2. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.

scholarly journals Preferences Single-Peaked on a Circle

2020 ◽  
Vol 68 ◽  
pp. 463-502 ◽  
Author(s):  
Dominik Peters ◽  
Martin Lackner
Keyword(s):  
Polynomial Time ◽  
Choice Theory ◽  
Social Choice Theory ◽  
Recognition Algorithm ◽  
Approval Voting ◽  
Voting Rules ◽  
One Dimensional ◽  
Facility Location Problems ◽  
Impossibility Results ◽  
Fast Recognition

We introduce the domain of preferences that are single-peaked on a circle, which is a generalization of the well-studied single-peaked domain. This preference restriction is useful, e.g., for scheduling decisions, certain facility location problems, and for one-dimensional decisions in the presence of extremist preferences. We give a fast recognition algorithm of this domain, provide a characterisation by finitely many forbidden subprofiles, and show that many popular single- and multi-winner voting rules are polynomial-time computable on this domain. In particular, we prove that Proportional Approval Voting can be computed in polynomial time for profiles that are single-peaked on a circle. In contrast, Kemeny's rule remains hard to evaluate, and several impossibility results from social choice theory can be proved using only profiles in this domain.

scholarly journals Notes on {a,b,c}-Modular Matrices

2021 ◽  
Author(s):  
Christoph Glanzer ◽  
Ingo Stallknecht ◽  
Robert Weismantel
Keyword(s):  
Polynomial Time ◽  
Standard Form ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Recognition Algorithm ◽  
Integral Matrix ◽  
Absolute Value ◽  
Integer Programs ◽  
Totally Unimodular Matrices

AbstractLet $A \in \mathbb {Z}^{m \times n}$ A ∈ ℤ m × n be an integral matrix and a, b, $c \in \mathbb {Z}$ c ∈ ℤ satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a,b,c}-modular, i.e., whether the set of n × n subdeterminants of A in absolute value is {a,b,c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k1 and k2 satisfying 2 ⋅|k1| = |k2|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a,b,c}-modular constraint matrices for any constants a, b and c.

scholarly journals Polynomial time recognition of vertices contained in all (or no) maximum dissociation sets of a tree

2021 ◽  
Vol 7 (1) ◽  
pp. 569-578
Author(s):  
Jianhua Tu ◽  
◽  
Lei Zhang ◽  
Junfeng Du ◽  
Rongling Lang ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Recognition Algorithm ◽  
Vertex Degree ◽  
Np Hard ◽  
Maximum Cardinality

<abstract><p>In a graph $ G $, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of the maximum dissociation set problem in various sub-classes of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and no maximum dissociation set of a tree. We present a linear time recognition algorithm which can determine whether a given vertex in a tree is contained in all (or no) maximum dissociation sets of the tree. Thus for a tree with $ n $ vertices, we can find all vertices belonging to all (or no) maximum dissociation sets of the tree in $ O(n^2) $ time.</p></abstract>

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