Papers by Iskandar Karapetyan
2017 Computer Science and Information Technologies (CSIT), 2017
Let D be a strongly connected balanced bipartite directed graph of order 2a ≥ 10 other than a dir... more Let D be a strongly connected balanced bipartite directed graph of order 2a ≥ 10 other than a directed cycle. Let x, y be distinct vertices in D. {x, y} dominates a vertex z if x → z and y → z; in this case, we call the pair {x, y} dominating. In this paper we prove: If max{d(x), d(y)} ≥ 2a − 2 for every dominating pair of vertices {x, y}, then D contains cycles of all lengths 2, 4,. .. , 2a − 2 or D is isomorphic to a certain digraph of order ten which we specify.
A Note on Short Paths in Oriented Graphs
Mathematical problems of computer science, Mar 25, 2010
On some Problems in Graph Theory
Mathematical problems of computer science, Dec 25, 2010
Cornell University - arXiv, Jul 24, 2012
Let D be a strong digraph on n ≥ 4 vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. B... more Let D be a strong digraph on n ≥ 4 vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) d(x) + d(y) ≥ 2n − 1 and min{d(x), d(y)} ≥ n − 1 for every pair of non-adjacent vertices x, y with a common in-neighbour or (**) min{d + (x) + d − (y), d − (x) + d + (y)} ≥ n for every pair of non-adjacent vertices x, y with a common in-neighbour or a common out-neighbour, then D is hamiltonian. In this paper we show that: (i) if D satisfies the condition (*) and the minimum semi-degree of D at least two or (ii) if D is not directed cycle and satisfies the condition (**), then either D contains a cycle of length n − 1 or n is even and D is isomorphic to complete bipartite digraph or to complete bipartite digraph minus one arc.
arXiv: Combinatorics, Apr 23, 2014
Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following conditi... more Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following condition (*): for every pair of non-adjacent vertices x, y with a common in-neighbour d(x) + d(y) ≥ 2n − 1 and min{d(x), d(y)} ≥ n − 1. In [2] (J. of Graph Theory 22 (2) (1996) 181-187)) J. Bang-Jensen, G. Gutin and H. Li proved that D is Hamiltonian. In [9] it was shown that if D satisfies the condition (*) and the minimum semi-degree of D at least two, then either D contains a pre-Hamiltonian cycle (i.e., a cycle of length n − 1) or n is even and D is isomorphic to the complete bipartite digraph (or to the complete bipartite digraph minus one arc) with partite sets of cardinalities of n/2 and n/2. In this paper we show that if the minimum out-degree of D at least two and the minimum in-degree of D at least three, then D contains also a Hamiltonian bypass, (i.e., a subdigraph is obtained from a Hamiltonian cycle by reversing exactly one arc).
Mathematical Problems of Computer Science, 2018
R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the fo... more R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. Problem. Let D be a strongly connected balanced bipartite directed graph of order 2a ≥8. Suppose that d(x) ≥ 2a - k, d(y) ≥a + k or d(y) ≥2a - k, d(x) ≥a + k for every pair of vertices {x; y}with a common out-neighbour, where 2 • k • a=2. Is D Hamiltonian? In this paper, we prove that if a digraph D satis¯es the conditions of this problem, then (i) D contains a cycle factor, (ii) for every vertex x ∈ V (D) there exists a vertex y ∈ V (D) such that x and y have a common out-neighbour.
Discrete Applied Mathematics, 2017
Let D be a strong digraph on n ≥ 4 vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. B... more Let D be a strong digraph on n ≥ 4 vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) d(x) + d(y) ≥ 2n − 1 and min{d(x), d(y)} ≥ n − 1 for every pair of non-adjacent vertices x, y with a common in-neighbour or (**) min{d + (x) + d − (y), d − (x) + d + (y)} ≥ n for every pair of non-adjacent vertices x, y with a common in-neighbour or a common out-neighbour, then D is hamiltonian. In this paper we show that: (i) if D satisfies the condition (*) and the minimum semi-degree of D at least two or (ii) if D is not directed cycle and satisfies the condition (**), then either D contains a cycle of length n − 1 or n is even and D is isomorphic to complete bipartite digraph or to complete bipartite digraph minus one arc.
Facta universitatis - series: Electronics and Energetics, 2007
Channel routing is an important phase of physical design of LSI and VLSI chips. The channel routi... more Channel routing is an important phase of physical design of LSI and VLSI chips. The channel routing method was first proposed by Akihiro Hashimoto and James Stevens [1]. The method was extensively studied by many authors and applied to different technologies. At present there are known many effective heuristic algorithms for channel routing. A. LaPaugh [2] proved that the restrictive routing problem is NP-complete. In this paper we prove that for every positive integer k there is a restrictive channel C for which ?(C)>? (HG)+L(VG)+k, where ? (C) is the thickness of the channel, ?(HG) is clique number of the horizontal constraints graph HG and L(VG) is the length of the longest directed path in the vertical constraints graph VG.
Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following conditi... more Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following condition (*): for every pair of non-adjacent vertices x, y with a common in-neighbour d(x) + d(y) ≥ 2n − 1 and min{d(x), d(y)} ≥ n − 1. In [2] (J. of Graph Theory 22 (2) (1996) 181-187)) J. Bang-Jensen, G. Gutin and H. Li proved that D is Hamiltonian. In [9] it was shown that if D satisfies the condition (*) and the minimum semi-degree of D at least two, then either D contains a pre-Hamiltonian cycle (i.e., a cycle of length n − 1) or n is even and D is isomorphic to the complete bipartite digraph (or to the complete bipartite digraph minus one arc) with partite sets of cardinalities of n/2 and n/2. In this paper we show that if the minimum out-degree of D at least two and the minimum in-degree of D at least three, then D contains also a Hamiltonian bypass, (i.e., a subdigraph is obtained from a Hamiltonian cycle by reversing exactly one arc).
Ninth International Conference on Computer Science and Information Technologies Revised Selected Papers, 2013
Let D be a strong digraph on n ≥ 4 vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J.... more Let D be a strong digraph on n ≥ 4 vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) d(x) + d(y) ≥ 2n − 1 and min{d + (x) + d − (y), d − (x) + d + (y)} ≥ n − 1 for every pair of non-adjacent vertices x, y with a common in-neighbour or a common out-neighbour, then D is hamiltonian. In this note we show that: if D is not directed cycle and satisfies the condition (*), then D contains a cycle of length n − 1 or n − 2.
Intersection Graphs of Rectangles and Segments
Lecture Notes in Computer Science, 2006
Let F be a finite family of sets and G(F) be the intersection graph of F (the vertices of G(F) ar... more Let F be a finite family of sets and G(F) be the intersection graph of F (the vertices of G(F) are the sets of family F and the edges of G(F) correspond to intersecting pairs of sets). The transversal number τ(F) is the minimum number of points meeting all sets of F. The independent (stability) number α(F) is the maximum number of pairwise disjoint sets in F. The clique number ω(F) is the maximum number of pairwise intersecting sets in F. The coloring number q(F) is the minimum number of classes in a partition of F into pairwise disjoint sets.
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Papers by Iskandar Karapetyan