A decomposition of a graph is a collection of edge-disjoint subgraphs 1 , 2 , . . . , of such that every edge of belongs to exactly one . If each is a path or a cycle in , then is called a path decomposition of . If each is a path in , then is called an acyclic path decomposition of . The minimum cardinality of a path decomposition (acyclic path decomposition) of is called the path decomposition number (acyclic path decomposition number) of and is denoted by ( ) ( ( )). In this paper we initiate a study of the parameter and determine the value of for some standard graphs. Further, we obtain some bounds for and characterize graphs attaining the bounds. We also prove that the difference between the parameters and can be made arbitrarily large.
Let G be a nontrivial, simple, finite, connected and undirected graph. A graphoidal decomposition (GD) of G is a collection ψ of paths and cycles in G that are internally disjoint such that every edge of G lies in exactly one member of ψ. As a variation of GD the notion of induced graphoidal decomposition (IGD) was introduced in [S. Arumugam, Path covers in graphs (2006)] which is a GD all of whose members are either induced paths or induced cycles. The minimum number of elements in such a decomposition of a graph G is called the IGD number, denoted by ηi(G). In this paper, we extend the study of the parameter ηi by establishing bounds for ηi(G) in terms of the diameter, girth and the maximum degree along with characterization of graphs achieving the bounds.
In a graph G = (V, E), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The minimum and maximum cardinalities of a maximal open packing set are respectively called the lower open packing number and the open packing number and are denoted by ρ o L and ρ o .In this paper, we present some bounds on these parameters.
A subset S of the vertex set V(G) of a graph G is called an isolate set if the subgraph induced by S has an isolated vertex. The subset S is called an isolate dominating set if it is both isolate and dominating. Also, S is called an isolate irredundant set if it is both isolate and irredundant. In this paper, we establish a chain connecting various isolate parameters with the existing domination parameters and discuss equality among the parameters in the extended chain.
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