کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4651552 | 1632578 | 2016 | 12 صفحه PDF | دانلود رایگان |
For a connected graph G=(V,E)G=(V,E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x−yx−y monophonic path is called an x−yx−ydetour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x−yx−y detour monophonic path for some elements x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G)dm(G). A detour monophonic set S of G is called a minimal detour monophonic set if no proper subset of S is a detour monophonic set of G. The upper detour monophonic number of G , denoted by dm+(G)dm+(G), is defined as the maximum cardinality of a minimal detour monophonic set of G . We determine bounds for it and find the upper detour monophonic number of certain classes of graphs. It is shown that for any three positive integers a,b,ca,b,c with 2≤a≤b≤c2≤a≤b≤c, there is a connected graph G with m(G)=a,dm(G)=bm(G)=a,dm(G)=b and dm+(G)=cdm+(G)=c. Also, for any three positive integers a,ba,b and n with 2≤a≤n≤b2≤a≤n≤b, there is a connected graph G with dm(G)=a,dm+(G)=bdm(G)=a,dm+(G)=b and a minimal detour monophonic set of cardinality n.
Journal: Electronic Notes in Discrete Mathematics - Volume 53, September 2016, Pages 331–342