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Waphare, B. N.
- On Removable and Non-Separating Even Cycles in Graphs
Abstract Views :210 |
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Authors
Y. M. Borse
1,
B. N. Waphare
1
Affiliations
1 Department of Mathematics, University of Pune, Pune 411007, IN
1 Department of Mathematics, University of Pune, Pune 411007, IN
Source
The Journal of the Indian Mathematical Society, Vol 80, No 3-4 (2013), Pagination: 221-234Abstract
Conlon proved that there exists an even cycle C in a 3-connected graph ≅K5, of minimum degree at least 4 such that G-E(C) is 2-connected and G - V(C) is connected. We prove that a 2-connected graph G of minimum degree at least 5 has an even cycle C such that G - E(C) is 2-connected and G - V(C) is connected.Keywords
2-Connected Graph, Even Cycle.
- Generalized Splitting Operation for Binary Matroids and its Applications
Abstract Views :193 |
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Authors
Affiliations
1 Department of Mathematics, University of Pune, Pune - 411 007, IN
2 Department of Mathematics, University of Urmia, Urmia 57135-165, IR
1 Department of Mathematics, University of Pune, Pune - 411 007, IN
2 Department of Mathematics, University of Urmia, Urmia 57135-165, IR
Source
The Journal of the Indian Mathematical Society, Vol 78, No 1-4 (2011), Pagination: 145-154Abstract
In this paper, we introduce the notion of generalized split- ting operation for binary matroids as an extension of the corresponding operation for graphs. The circuits and the bases of the new matroid are characterized. Under the reverse operation, we determine those binary matroids, each of which yields a given matroid by applying the splitting operation on them.Keywords
Graph, Binary Matroid, Circuit, Minor, Splitting Operation.- On Removable Cycles in Connected Graphs
Abstract Views :205 |
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Authors
Y. M. Borse
1,
B. N. Waphare
1
Affiliations
1 Department of Mathematics, University of Pune, Pune 411 007, IN
1 Department of Mathematics, University of Pune, Pune 411 007, IN
Source
The Journal of the Indian Mathematical Society, Vol 76, No 1-4 (2009), Pagination: 31-46Abstract
We call a cycle C of a graph G removable if G - E(C) is connected. In this paper, we obtain sufficient conditions for the existence of a removable cycle in a connected graph G which is edge-disjoint from a connected subgraph of G. Also, a characterization of connected graphs of minimum degree at least 3 having two edge-disjoint removable cycles is obtained in terms of forbidden graphs. We provide sufficient conditions for the existence of removable even cycles, and also for the existence of odd cycles. Further, we handle the problem of determining when a given edge of a connected graph can be guaranteed to lie in some removable cycle.Keywords
Removable Cycle, Connected Graph.- Modular Pairs, Standard Elements, Neutral Elements and Related Results in Partially Ordered Sets
Abstract Views :193 |
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Authors
Affiliations
1 4B, Mitrakunj Society, Sakri Road, Dhule-424 001, IN
2 Department of Mathematics, S.S.V.P.S’s. L.K. Dr. PR. Ghogrey Science College, Dhule-424 005, IN
3 Department of Mathematics, University of Pune, Pune 411 007, IN
1 4B, Mitrakunj Society, Sakri Road, Dhule-424 001, IN
2 Department of Mathematics, S.S.V.P.S’s. L.K. Dr. PR. Ghogrey Science College, Dhule-424 005, IN
3 Department of Mathematics, University of Pune, Pune 411 007, IN
Source
The Journal of the Indian Mathematical Society, Vol 71, No 1-4 (2004), Pagination: 13-53Abstract
In an attempt to answer the open problem of Birkhoff, namely, “How should one define modular pairs in a general poset?”, five distinct notions of modular pair in a general poset are discussed and studied. Interrelationships between these five concepts are looked into and several counter-examples are supplied. The covering properties and the exchange property in general posets are thoroughly studied and several characterizations of these properties are obtained. As an offshoot of our study, twentythree characterizations of the covering property in atomistic lattices are accomplished.
Investigations of del-relation, del-tilda relation, perspectivity, sub-perspectivity etc. in posets are carried out.
In the context of another open problem posed by Birkhoff concerning how to define natural extensions of the concepts of neutral elements and standard elements we cover substantial ground and supply adequate solution to this query. Special posets such as SSC-poseLs, SSC*-posets, orthomodular and orthocomplemented posets are studied and several characterizations are obtained.
Keywords
Modular Pair, Covering Property, Exchange Property, Del-Relation, Del-Tilda Relation, Perspectivity, Subperspectivity, g-Atomic Poset, g-Atomistic Poset, SSC-Poset, SSC*-Poset, Standard Element, Neutral Element, Central Element, g-Central Element.- Modular Pairs, Covering Property and Related Results in Posets
Abstract Views :133 |
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Authors
Affiliations
1 13, General Arun Kumar Vaidya Nagar, Off Sakri Road, Dhule-424001, IN
2 Iwaidani 6-333-10, Matsuyama 790, JP
3 Department of Mathematics, University of Pune, Pune-411007, IN
1 13, General Arun Kumar Vaidya Nagar, Off Sakri Road, Dhule-424001, IN
2 Iwaidani 6-333-10, Matsuyama 790, JP
3 Department of Mathematics, University of Pune, Pune-411007, IN
Source
The Journal of the Indian Mathematical Society, Vol 70, No 1-4 (2003), Pagination: 229-253Abstract
How should one define a modular pair in a general poset? queried Birkhoff, sixty years ago. Not only we answer this open problem satisfactorily but obtain interesting properties concerning covering property, exchange property in a poset with zero. In this context a few counter examples are also supplied. This general study has led us, as an offshoot, to thirteen characterizations of covering property in a lattice with zero. Del-relation and perspectivity are characterized in posets. The study of atom spaces is extended to posets. Statischness in atomistic posets is characterized. Further, ortho-modular posets are also characterized and an interesting open problem in this context is raised.Keywords
Poset, Upper (Lower) Cone, Semilattice, Lattice, Covering Relation, Atom, Modular Pair, Covering Property, Exchange Property, Del-Relation, Perspectivity, Atom Space, Orthomodular Posets.- Posets Dismantlable by Doubly Irreducibles
Abstract Views :199 |
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Authors
Affiliations
1 Department of Mathematics, P.E.S. Modern College of Arts, Science and Commerce, Shivajinagar, Pune-411005, IN
2 Centre for Advanced Study in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune-411007, IN
1 Department of Mathematics, P.E.S. Modern College of Arts, Science and Commerce, Shivajinagar, Pune-411005, IN
2 Centre for Advanced Study in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune-411007, IN
Source
The Journal of the Indian Mathematical Society, Vol 88, No 1-2 (2021), Pagination: 46–59Abstract
In this paper, we introduce the concept of a poset dismantlable by doubly irreducibles. We also introduce the operations, `1-sum' and `2-sum' of posets. Using these operations, we obtain the structure theorem for posets dismantlable by doubly irreducibles, which generalizes the structure theorem for dismantlable lattices.Keywords
Chain, Lattice, Poset, Doubly irreducible elementReferences
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