The induced path number of the complements of some graphs (original) (raw)

Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement

Acta Mathematica Sinica, English Series, 2012

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. Broere et al. proved that if G is a graph of order n, then √ n ≤ ρ(G) + ρ(G) ≤ 3n 2. In this paper, we characterize the graphs G for which ρ(G) + ρ(G) = 3n 2 , improve the lower bound on ρ(G) + ρ(G) by one when n is the square of an odd integer, and determine a best possible upper bound for ρ(G) + ρ(G) when neither G nor G has isolated vertices.

Path Decomposition Number of Certain Graphs

Open Journal of Discrete Applied Mathematics, 2018

Let G be a simple, finite and connected graph. A graph is said to be decomposed into subgraphs H 1 and H 2 which is denoted by G = H 1 ⊕ H 2 , if G is the edge disjoint union of H 1 and H 2. Assume that G = H 1 ⊕ H 2 ⊕ • • • ⊕ H k and if each H i , 1 ≤ i ≤ k, is a path or cycle in G, then the collection of edge-disjoint subgraphs of G denoted by ψ is called a path decomposition of G. If each H i is a path in G then ψ is called an acyclic path decomposition of G. The minimum cardinality of a path decomposition of G, denoted by π(G), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of G, denoted by πa(G), is called the acyclic path decomposition number of G. In this paper, we determine path decomposition number for a number of graphs in particular, the Cartesian product of graphs. We also provided bounds for π(G) and πa(G) for these graphs.

On the -path vertex cover of some graph products

Discrete Mathematics, 2013

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψ k (G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for ψ k of the Cartesian and the direct product of paths are derived. It is shown that for ψ 3 those bounds are tight. For the lexicographic product bounds are presented for ψ k , moreover ψ 2 and ψ 3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.

On the Generalized Complement of Some Graphs

Asia Pacific Journal of Mathematics, 2021

A. In this paper we study the generalized complement of the graph G m,n = (V, E) for some values of m, n. We study the generalized complement of G m,n graphs with respect to the equal degree partition. The 2−complement of G m,n graphs are also determined for m = 2, n is even or odd. In particular, for some values of m, n ∈ N, we studied the complement of G m,n graphs with respect to the equal degree partition and the 2−complement of G m,n graphs. We determine the partitions P k , k ∈ N of the vertex set V such that the generalized complement of G m,n graph is a path graph and a comb graph.

A survey of the Path Partition Conjecture

Discussiones Mathematicae Graph Theory, 2013

The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2) any pair of positive integers such that G has no path with more than λ 1 + λ 2 vertices, then there exists a partition (V 1 , V 2) of the vertex set of G such that V i has no path with more than λ i vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the PPC that have appeared in the literature since its first formulation in 1981.

A study on the pendant number of graph products

Acta Universitatis Sapientiae, Informatica

A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

On the path separation number of graphs

A path separator of a graph G is a set of paths P = {P 1 , . . . , Pt} such that for every pair of edges e, f ∈ E(G), there exist paths Pe, P f ∈ P such that e ∈ E(Pe), f ∈ E(Pe), e ∈ E(P f ) and f ∈ E(P f ). The path separation number of G, denoted psn(G), is the smallest number of paths in a path separator. We shall estimate the path separation number of several graph families, including complete graphs, random graph, the hypercube, and discuss general graphs as well.

C O ] 2 J un 2 01 6 On the path separation number of graphs

2017

A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of edges e, f ∈ E(G), there exist paths Pe, Pf ∈ P such that e ∈ E(Pe), f 6∈ E(Pe), e 6∈ E(Pf ) and f ∈ E(Pf ). The path separation number of G, denoted psn(G), is the smallest number of paths in a path separator. We shall estimate the path separation number of several graph families – including complete graphs, random graph, the hypercube – and discuss general graphs as well.

Graph partition into paths containing specified vertices

Discrete Mathematics, 2002

For a graph G, let 2(G) denote the minimum degree sum of a pair of nonadjacent vertices. Suppose G is a graph of order n. Enomoto and Ota (J. Graph Theory 34 (2000) 163-169) conjectured that, if a partition n = k i=1 ai is given and 2(G) ¿ n + k − 1, then for any k distinct vertices v1; : : : ; v k , G can be decomposed into vertex-disjoint paths P1; : : : ; P k such that |V (Pi)| = ai and vi is an endvertex of Pi. Enomoto and Ota (J. Graph Theory 34 (2000) 163) veriÿed the conjecture for the case where all ai 6 5, and the case where k 6 3. In this paper, we prove the following theorem, with a stronger assumption of the conjecture. Suppose G is a graph of order n. If a partition n = k i=1 ai is given and 2(G) ¿ k i=1 max(4 3 ai ; ai + 1) − 1, then for any k distinct vertices v1; : : : ; v k , G can be decomposed into vertex-disjoint paths P1; : : : ; P k such that |V (Pi)| = ai and vi is an endvertex of Pi for all i. This theorem implies that the conjecture is true for the case where all ai 6 5 which was proved in (J. Graph Theory 34 (2000) 163-169).