Abstract
A restrained Steiner set of a connected graph 𝑮 of order 𝒑≥𝟐 is a set 𝑾⊆𝑽(𝑮)such that 𝑾 is a Steiner set, and if either 𝑾 = 𝑽 or the subgraph𝑮[𝑽−𝑾] inducedby [𝑽−𝑾] has no isolated vertices. The restrained Steiner number 𝒔𝒓(𝑮) of 𝑮 isthe minimum cardinality of its restrained Steiner sets and any restrained Steinerset of cardinality 𝒔𝒓(𝑮) is a minimum restrained Steiner set of 𝑮. For a minimum restrained Steiner set 𝑾of 𝑮, a subset 𝑻⊆𝑾 is called a forcing subset for 𝑾 if 𝑾is the unique minimum restrained Steiner set containing 𝑻. A forcing subset for 𝑾of minimum cardinality is a minimum forcing subset of 𝑾. The forcing restrained Steiner number of 𝑾, denoted by 𝒇𝒓𝒔(𝑾), is the cardinality of a minimum forcingsubset of 𝑾. The forcing restrained Steiner number of 𝑮, denoted by 𝒇𝒓𝒔(𝑮)is𝒇𝒓𝒔(𝑮)=𝒎𝒊𝒏 {𝒇𝒓𝒔(𝑾)}, where the minimum is taken over all minimum restrainedSteiner sets 𝑾 in 𝑮. Some general properties satisfied by the concept forcing restrained Steiner number are studied. The forcing restrained Steiner number of certain classes of graphs is determined. It is shown that for every pair 𝒂,𝒃 ofintegers with 𝟎≤𝒂 < 𝒃and 𝒃≥𝟐, there exists a connected graph 𝑮 such that𝒇𝒓𝒔(𝑮)=𝒂 and 𝒔𝒓(𝑮)=𝒃.
Published Version
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