Some Steiner concepts on lexicographic products of graphs

Let G be a graph and W a subset of V(G). A subtree with the minimum number of edges that contains all vertices of W is a Steiner tree for W. The number of edges of such a tree is the Steiner distance of W and union of all vertices belonging to Steiner trees for W form a Steiner interval. We describe both of these for the lexicographic product of graphs. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number.

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Two open problems in the fixed point theory of contractive type mappings on quasimetric spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2017.02.04 ◽

2017 ◽

Vol 33 (2) ◽

pp. 169-180

Author(s):

MITROFAN M. CHOBAN ◽

◽

VASILE BERINDE ◽

◽

Keyword(s):

Fixed Point ◽

Metric Spaces ◽

Fixed Point Theory ◽

Point Theory ◽

Partial Answer ◽

Open Problems ◽

Complete Answer ◽

Contractive Type ◽

Generalized Distances

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22 (2013), No. 1, 23–32] are considered. We give a complete answer to the first problem, a partial answer to the second one, and also illustrate the complexity and relevance of these problems by means of four very interesting and comprehensive examples.

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Another H-super magic decompositions of the lexicographic product of graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.2 ◽

2018 ◽

Vol 2 (2) ◽

pp. 72

Author(s):

H Hendy ◽

Kiki A. Sugeng ◽

A.N.M Salman ◽

Nisa Ayunda

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Graph Decomposition ◽

Graph Labeling ◽

Sufficient Condition ◽

Lexicographic Product ◽

Simple Graphs ◽

Product Of Graphs

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2, denoted by G1[G2], is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn, n where n is the order of G2. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where n > 3, m > 1, and t = 3, 4, n − 2.

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Approximating Steiner Trees and Forests with Minimum Number of Steiner Points

Approximation and Online Algorithms - Lecture Notes in Computer Science ◽

10.1007/978-3-319-18263-6_9 ◽

2015 ◽

pp. 95-106

Author(s):

Nachshon Cohen ◽

Zeev Nutov

Keyword(s):

Steiner Trees ◽

Steiner Points ◽

Minimum Number

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Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽

10.3390/a11100161 ◽

2018 ◽

Vol 11 (10) ◽

pp. 161 ◽

Cited By ~ 1

Author(s):

R. Vignesh ◽

J. Geetha ◽

K. Somasundaram

Keyword(s):

Chromatic Number ◽

Maximum Degree ◽

Line Graph ◽

Product Line ◽

Total Coloring ◽

Lexicographic Product ◽

Total Chromatic Number ◽

Minimum Number ◽

Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

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2-Distance chromatic number of some graph products

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500214 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050021

Author(s):

Ghazale Ghazi ◽

Freydoon Rahbarnia ◽

Mostafa Tavakoli

Keyword(s):

Chromatic Number ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Graph Products ◽

Lexicographic Product ◽

Graph Product ◽

Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

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Doubly connected domination in the corona and lexicographic product of graphs

Applied Mathematical Sciences ◽

10.12988/ams.2014.4136 ◽

2014 ◽

Vol 8 ◽

pp. 1521-1533 ◽

Cited By ~ 1

Author(s):

Benjier H. Arriola ◽

Sergio R. Canoy, Jr.

Keyword(s):

Lexicographic Product ◽

Product Of Graphs

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b -continuity and the lexicographic product of graphs

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2015.07.024 ◽

2015 ◽

Vol 50 ◽

pp. 139-144 ◽

Cited By ~ 2

Author(s):

Cláudia Linhares Sales ◽

Rafael Vargas ◽

Leonardo Sampaio

Keyword(s):

Lexicographic Product ◽

Product Of Graphs

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Univalence of a complex linear combination of two extremal parallel slit mappings

Analysis ◽

10.1524/anly.2007.27.2-3.301 ◽

2007 ◽

Vol 27 (2-3) ◽

Cited By ~ 2

Author(s):

Masakazu Shiba

Keyword(s):

Linear Combination ◽

Present Article ◽

Convex Combination ◽

Plane Domain ◽

Partial Answer ◽

Vertical Slit ◽

Complete Answer ◽

Conformal Welding ◽

Parallel Slit ◽

Complex Linear

Any convex combination of the so-called extremal horizontal and vertical slit mappings of a plane domain is known to be univalent. In his study on the conformal welding of annuli F. Maitani needed to know when a complex linear combination is univalent and he gave a partial answer to this problem. In the present article we give a complete answer to his and to generalized problems.

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The geodetic number of the lexicographic product of graphs

Discrete Mathematics ◽

10.1016/j.disc.2011.04.004 ◽

2011 ◽

Vol 311 (16) ◽

pp. 1693-1698 ◽

Cited By ~ 13

Author(s):

Boštjan Brešar ◽

Tadeja Kraner Šumenjak ◽

Aleksandra Tepeh

Keyword(s):

Lexicographic Product ◽

Geodetic Number ◽

Product Of Graphs

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Identifying Codes of Lexicographic Product of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2974 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 8

Author(s):

Min Feng ◽

Min Xu ◽

Kaishun Wang

Keyword(s):

Connected Graph ◽

Necessary Condition ◽

Lexicographic Product ◽

Arbitrary Graph ◽

Minimum Cardinality ◽

Identifying Codes ◽

Two Parameters ◽

Sufficient And Necessary Condition ◽

Product Of Graphs

Let $G$ be a connected graph and $H$ be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product $G[H]$ of $G$ and $H$. We first introduce two parameters of $H$, which are closely related to identifying codes of $H$. Then we provide the sufficient and necessary condition for $G[H]$ to be identifiable. Finally, if $G[H]$ is identifiable, we determine the minimum cardinality of identifying codes of $G[H]$ in terms of the order of $G$ and these two parameters of $H$.

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