The Forcing Restrained Steiner Number of a Graph
M. S. Malchijah Raj1, J. John2
1M. S. Malchijah Raj, Scholar, Research and Development Centre, Bharathiar University, Coimbatore (Tamil Nadu), India.
2J. John, Department of Mathematics, Government College of Engineering, Tirunelveli (Tamil Nadu), India.
Manuscript received on 30 September 2019 | Revised Manuscript received on 12 November 2019 | Manuscript Published on 22 November 2019 | PP: 1799-1803 | Volume-8 Issue-6S3 September 2019 | Retrieval Number: F13820986S319/19ยฉBEIESP | DOI: 10.35940/ijeat.F1382.0986S319
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Abstract: A restrained Steiner set of a connected graph ๐‘ฎ of order ๐’‘ โ‰ฅ ๐Ÿ is a set ๐‘พ โŠ† ๐‘ฝ(๐‘ฎ)such that ๐‘พ is a Steiner set, and if either ๐‘พ = ๐‘ฝ or the subgraph๐‘ฎ[๐‘ฝ โˆ’ ๐‘พ] inducedby [๐‘ฝ โˆ’ ๐‘พ] has no isolated vertices. The restrained Steiner number ๐’”๐’“ ๐‘ฎ of ๐‘ฎ isthe minimum cardinality of its restrained Steiner sets and any restrained Steinerset of cardinality ๐’”๐’“ ๐‘ฎ is a minimum restrained Steiner set of ๐‘ฎ. For a minimum restrained Steiner set ๐‘พof ๐‘ฎ, a subset ๐‘ป โŠ† ๐‘พ is called a forcing subset for ๐‘พ if ๐‘พis the unique minimum restrained Steiner set containing ๐‘ป. A forcing subset for ๐‘พof minimum cardinality is a minimum forcing subset of ๐‘พ. The forcing restrained Steiner number of ๐‘พ, denoted by ๐’‡๐’“๐’” ๐‘พ , is the cardinality of a minimum forcingsubset of ๐‘พ. The forcing restrained Steiner number of ๐‘ฎ, denoted by ๐’‡๐’“๐’” ๐‘ฎ is๐’‡๐’“๐’” ๐‘ฎ = ๐’Ž๐’Š๐’โก{๐’‡๐’“๐’” ๐‘พ }, where the minimum is taken over all minimum restrainedSteiner sets ๐‘พ in ๐‘ฎ. 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 ๐’‚,๐’ƒ ofintegers with ๐ŸŽ โ‰ค ๐’‚ < ๐’ƒand ๐’ƒ โ‰ฅ ๐Ÿ, there exists a connected graph ๐‘ฎ such that๐’‡๐’“๐’” ๐‘ฎ = ๐’‚ and ๐’”๐’“ ๐‘ฎ = ๐’ƒ.
Keywords: Steiner Distance, Steiner Number, Forcing Steiner Number, Restrained, Steiner Number, Forcing Restrained Steiner Number.
Scope of the Article: Cryptography and Applied Mathematics