engleski

K-domination sets on some extremal hexagonal chains

Hexagonal systems are geometric objects obtained by arranging mutually congruent regular hexagons in the plane. They are of considerable importance in theoretical chemistry because they are natural graph representation of benzenoid hydrocarbons. A hexagonal chain is a catacondensed hexagonal system in which every hexagon is adjacent to at most two hexagons. In chemistry important topological invariants such as Hosoya index, Merrifield Simmons index and the largest eigenvalue, were investigated on hexagonal chains, and it has been shown that there are three special types of hexagonal chains which are extremal ones due to these invariants. Those are: the linear chain L_h, the zig-zag chain Z_h and the helicence chain H_h, where h is the number of hexagons in a chain. For any graph G by V(G) and E(G) we denote the vertex-set and the edge-set of G, respectively. For graph G subset D of the vertex-set of G is called k-dominating set, k ≥ 1, if for every vertex v ∈ V (G) \D, there exists at least one vertex w ∈ D, such that d(v, w) ≤ k. The k-domination number k(G) is the cardinality of the smallest k-dominating set. The 1-domination set (number) is also called domination set (number). K-dominating sets and k-domination number were already determined for L_h. In this paper we determine k-dominating sets and upper bounds for k-domination number for Z_h and C_h.

k-dominating set; k-domination number; linear hexagonal chain; zig-zag chain; helicence chain.

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