Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON PATHOS BLOCK LINE CUT-VERTEX GRAPH OF A TREE

  • Received : 2017.10.28
  • Accepted : 2019.10.31
  • Published : 2020.01.31
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Abstract

A pathos block line cut-vertex graph of a tree T, written P BLc(T), is a graph whose vertices are the blocks, cut-vertices, and paths of a pathos of T, with two vertices of P BLc(T) adjacent whenever the corresponding blocks of T have a vertex in common or the edge lies on the corresponding path of the pathos or one corresponds to a block Bi of T and the other corresponds to a cut-vertex cj of T such that cj is in Bi; two distinct pathos vertices Pm and Pn of P BLc(T) are adjacent whenever the corresponding paths of the pathos Pm(vi, vj) and Pn(vk, vl) have a common vertex. We study the properties of P BLc(T) and present the characterization of graphs whose P BLc(T) are planar; outerplanar; maximal outerplanar; minimally nonouterplanar; eulerian; and hamiltonian. We further show that for any tree T, the crossing number of P BLc(T) can never be one.

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References

  1. F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, MA, 1969.
  2. F. Harary, Covering and packing in graphs. I, Ann. New York Acad. Sci. 175 (1970), 198-205. https://doi.org/10.1111/j.1749-6632.1970.tb56470.x
  3. F. Harary and G. Prins, The block-cutpoint-tree of a graph, Publ. Math. Debrecen 13 (1966), 103-107.
  4. V. R. Kulli, On minimally nonouterplanar graphs, Proc. Indian Nat. Sci. Acad. Part A 41 (1975), no. 3, 275-280.
  5. V. R. Kulli, A characterization of paths, Math. Ed. 9 (1975), A1-A2.
  6. V. R. Kulli and M. H. Muddebihal, On lict and litact graph of a graph, Proceeding of the Indian National Science Academy. 41 (1975), 275-280.
  7. M. H. Muddebihal, B. R. Gudagudi, and R. Chandrasekhar, On pathos line graph of a tree, Nat. Acad. Sci. Lett. 24 (2001), no. 5-12, 116-123.