Abstract
The axiomatic approach with the interval function, induced path transit function and all-paths transit function of a connected graph form a well studied area in metric and related graph theory. In this paper we introduce the first order axiom:
(cp) For any pairwise distinct vertices \(a,b,c,d\in V\)
\(b \in R(a,c)\) and \(b \in R(a,d)\Rightarrow c\in R(b,d)\) or \(d\in R(b,c)\).
We study this new axiom on the interval function, induced path transit function and all-paths transit function of a connected simple and finite graph. We present characterizations of claw and paw-free graphs using this axiom on standard path transit functions on graphs, namely the interval function, induced path transit function and the all-paths transit function. The family of 2-connected graphs for which the axiom (cp) is satisfied on the interval function and the induced path transit function are Hamiltonian. Additionally, we study arbitrary transit functions whose underlying graphs are Hamiltonian.
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References
Balakrishnan, K., Changat, M., Lakshmikuttyamma, A.K., Mathews, J., Mulder, H.M., Narasimha-Shenoi, P.G., Narayanan, N.: Axiomatic characterization of the interval function of a block graph. Disc. Math. 338, 885–894 (2015)
Changat, M., Klavžar, S., Mulder, H.M.: The all-paths transit function of a graph. Czech. Math. J. 51(126), 439–448 (2001)
Changat, M., Mathew, J.: Induced path transit function, monotone and Peano axioms. Disc. Math. 286(3), 185–194 (2004)
Changat, M., Mulder, H.M., Sierksma, G.: Convexities related to path properties on graphs. Disc. Math. 290(2–3), 117–131 (2005)
Changat, M., Mathews, J., Mulder, H.M.: The induced path function, monotonicity and betweenness. Disc. Appl. Math. 158(5), 426–433 (2010)
Changat, M., Lakshmikuttyamma, A.K., Mathews, J., Peterin, I., Narasimha-Shenoi, P.G., Seethakuttyamma, G., Spacapan, S.: A forbiddensubgraph characterization of some graph classes using betweenness axioms. Disc. Math. 313, 951–958 (2013)
Duchet, P.: Convex sets in graphsII. Minimal path convexity. J. Combin. Theory Ser. B. 44, 307–316 (1988). (1984)
Goodman, S., Hedetniemi, S.: Sufficient conditions for agraph to be Hamiltonian. J. Combin. Theory Ser. B 16, 175–180 (1974)
Gould, R.J., Jacobson, M.S.: Forbidden subgraphs and Hamiltonian properties of graphs. Disc. Math. 42(2), 189–196 (1982)
Morgana, M.A., Mulder, H.M.: The induced path convexity, betweenness and svelte graphs. Disc. Math. 254, 349–370 (2002)
Mulder, H.M.: The Interval function of a Graph. MC Tract 132, Mathematisch Centrum, Amsterdam (1980)
Mulder, H.M.: Transit functions on graphs (and posets). In: Changat, M., Klavžar, S., Mulder, H.M., Vijayakumar, A. (eds.) Convexity in Discrete Structures. Lecture Notes Series, pp. 117–130. Ramanujan Math. Soc., Mysore (2008)
Mulder, H.M., Nebeský, L.: Axiomatic characterization of the interval function of a graph. European J. Combin. 30, 1172–1185 (2009)
Nebeský, L.: A characterization of the interval function of a connected graph. Czech. Math. J. 44, 173–178 (1994)
Nebeský, L.: Characterizing the interval function of a connected graph. Math. Bohem. 123(2), 137–144 (1998)
Nebeský, L.: Characterization of the interval function of a (finite or infinite) connected graph. Czech. Math. J. 51, 635–642 (2001)
Nebeský, L.: The induced paths in a connected graph and a ternary relation determined by them. Math. Bohem. 127, 397–408 (2002)
Acknowledgments
This research work is supported by NBHM-DAE, Govt. of India under grantNo. 2/48(9)/2014/ NBHM(R.P)/R& D-II/4364 DATED 17TH NOV, 2014.
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Changat, M., Hossein Nezhad, F., Narayanan, N. (2016). Axiomatic Characterization of Claw and Paw-Free Graphs Using Graph Transit Functions. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_10
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DOI: https://doi.org/10.1007/978-3-319-29221-2_10
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