Nordhaus–Gaddum Results for the Induced Path Number of a Graph When Neither the Graph Nor Its Complement Contains Isolates

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.

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Path number of bipartite digraphs

Journal of Combinatorial Theory Series B ◽

10.1016/0095-8956(80)90068-4 ◽

1980 ◽

Vol 28 (2) ◽

pp. 243-244 ◽

Cited By ~ 3

Author(s):

William G Frye ◽

Renu Laskar

Keyword(s):

Path Number

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A Study on NC Gear Cutting of Large-size Bevel Gears : 3rd Report; A Method to Save Path Number of the Cutter in Tooth Generation

Bulletin of JSME ◽

10.1299/jsme1958.28.1301 ◽

1985 ◽

Vol 28 (240) ◽

pp. 1301-1307

Author(s):

Shigeyuki SHIMACHI ◽

Takao SAKAI ◽

Takashi EMURA

Keyword(s):

Bevel Gears ◽

Gear Cutting ◽

Large Size ◽

Path Number

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Edge lifting and Roman domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500646 ◽

2020 ◽

pp. 2150064

Author(s):

Hicham Meraimi ◽

Mustapha Chellali

Keyword(s):

Domination Number ◽

Roman Domination ◽

Roman Domination Number ◽

Induced Path ◽

Domination In Graphs

Let [Formula: see text] be a graph, and let [Formula: see text] be an induced path centered at [Formula: see text]. An edge lift defined on [Formula: see text] is the action of removing edges [Formula: see text] and [Formula: see text] while adding the edge [Formula: see text] to the edge set of [Formula: see text]. In this paper, we initiate the study of the effects of edge lifting on the Roman domination number of a graph, where various properties are established. A characterization of all trees for which every edge lift increases the Roman domination number is provided. Moreover, we characterize the edge lift of a graph decreasing the Roman domination number, and we show that there are no graphs with at most one cycle for which every possible edge lift can have this property.

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Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

Discrete Dynamics in Nature and Society ◽

10.1155/2019/8696982 ◽

2019 ◽

Vol 2019 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Lian Chen ◽

Abid Mehboob ◽

Haseeb Ahmad ◽

Waqas Nazeer ◽

Muhammad Hussain ◽

...

Keyword(s):

Topological Index ◽

Molecular Descriptor ◽

Wiener Index ◽

Vital Role ◽

Harary Index ◽

Chemical Graph ◽

Chemical Graph Theory ◽

Hosoya Polynomial ◽

Molecular Branching ◽

Path Number

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.

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