scholarly journals Quasi-Eulerian Hypergraphs

10.37236/6361 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Amin Bahmanian ◽  
Mateja Šajna
Keyword(s):  
Sufficient Conditions ◽  
Single Component ◽  
Necessary And Sufficient Conditions ◽  
Uniform Hypergraph ◽  
Triple Systems ◽  
Universal Cycles ◽  
Np Complete ◽  
Euler Tour ◽  
Necessary And Sufficient ◽  
Overlap Cycles

We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tourthus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in particular, we show that every 3-uniform hypergraph without cut edges admits an Euler family. Finally, we show that the problem of existence of an Euler family is polynomial on the class of all hypergraphs.This work complements existing results on rank-1 universal cycles and 1-overlap cycles in triple systems, as well as recent results by Lonc and Naroski, who showed that the problem of existence of an Euler tour in a hypergraph is NP-complete.

APPROXIMATING THE NEAREST NEIGHBOR INTERCHARGE DISTANCE FOR NON-UNIFORM-DEGREE EVOLUTIONARY TREES

2001 ◽  
Vol 12 (04) ◽  
pp. 533-550 ◽  
Author(s):  
WING-KAI HON ◽  
TAK-WAH LAM
Keyword(s):  
Approximation Algorithms ◽  
Maximum Degree ◽  
Nearest Neighbor ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Approximation Ratio ◽  
Evolutionary Trees ◽  
Weighted Trees ◽  
Np Complete ◽  
Necessary And Sufficient

The nearest neighbor interchange (nni) distance is a classical metric for measuring the distance (dissimilarity) between evolutionary trees. It has been known that computing the nni distance is NP-complete. Existing approximation algorithms can attain an approximation ratio log n for unweighted trees and 4 log n for weighted trees; yet these algorithms are limited to degree-3 trees. This paper extends the study of nni distance to trees with non-uniform degrees. We formulate the necessary and sufficient conditions for nni transformation and devise more topology-sensitive approximation algorithms to handle trees with non-uniform degrees. The approximation ratios are respectively [Formula: see text] and [Formula: see text] for unweighted and weighted trees, where d ≥ 4 is the maximum degree of the input trees.

scholarly journals Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals Embedding Factorizations for 3-Uniform Hypergraphs II: $r$-Factorizations into $s$-Factorizations

10.37236/5714 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Amin Bahmanian ◽  
Mike Newman
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Necessary And Sufficient ◽  
Existing Result

Motivated by a 40-year-old problem due to Peter Cameron on extending partial parallelisms, we provide necessary and sufficient conditions under which one can extend an $r$-factorization of a complete $3$-uniform hypergraph on $m$ vertices, $K_m^3$, to an $s$-factorization of $K_n^3$. This generalizes an existing result of Baranyai and Brouwer — where they proved it for the case $r=s=1$.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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