Yashwant Manikrao Borse

• M.Sc. M.Phil. Ph.D.
• Professor at Savitribai Phule Pune University

The paired, total, and independent domination subdivision number of a graph $G$ is the minimum number of edges that must be subdivided, where each edge can be subdivided at most once, in order to increase the paired, total, and independent domination number, respectively. In this paper, we prove that the corresponding decision problems for paired,...

In this paper, we offer a simple combinatorial approach for calculating the distance matrix, distance-based structural polynomials and many distance-based indices of the zero-divisor graph [Formula: see text] associated to a finite reduced ring [Formula: see text] where [Formula: see text] is a finite field. A key contribution is the development of...

For an integer \(k \ge 3,\) the sunlet graph of order 2k, denoted by \(L_{2k},\) is a graph obtained from a cycle of length k by attaching one pendant vertex to each vertex of the cycle. The cycle decompositions of the hypercubes is well studied in the literature. In this paper, we obtain sunlet decompositions of hypercubes using cycle decompositio...

Let Rn be a finite reduced ring with n maximal ideals 𝔪i and let Γ(Rn) be the zero-divisor graph associated to Rn. The class of rings Rn contains the Boolean rings as a subclass. When Rn/𝔪i = 𝔽 for all i where 𝔽 is a finite field, we associate two (n − 1) × (n − 1) sized matrices P and Q to the graph Γ(Rn) having combinatorial entries and use these...

An elementary lift of a binary matroid M that arises from a binary coextension of M can easily be obtained by applying the splitting operation on M. This operation on a graphic matroid may not produce a graphic matroid. We give a method to determine the forbidden minors for the class of graphic matroids M such that the splitting of M by any set of...

Let [Formula: see text] be a connected graph with minimum degree at least [Formula: see text] and let [Formula: see text] be an integer such that [Formula: see text] The conditional [Formula: see text]-edge ([Formula: see text]-vertex) cut of [Formula: see text] is defined as a set [Formula: see text] of edges (vertices) of [Formula: see text] whos...

The sunlet graph of order eight, denoted by [Formula: see text] is the graph obtained by adding a pendant edge to each vertex of a cycle of length [Formula: see text] We prove that the [Formula: see text]-dimensional hypercube [Formula: see text] can be decomposed into the copies of the graph [Formula: see text] if and only if [Formula: see text] i...

The augmented cube AQ n is one of the important variations of the hypercube Q n . In this paper, we prove that the conditional h-edge connectivity of AQ n with n ≥ 3 is 8n − 16 for h = 3 and 2 ⁿ for h = 2n − 3. We also obtain an upper bound on the conditional h-edge connectivity for odd integer h satisfying [Formula: see text].

The n-dimensional augmented cube AQn is a variation of the hypercube It is a -regular and -connected graph on vertices. One of the fundamental properties of AQn is that it is pancyclic, that is, it contains a cycle of every length from 3 to In this paper, we generalize this property to k-regular subgraphs for and We prove that the augmented cube AQ...

In this paper, we consider the problem of decomposing the augmented cube AQn into two spanning, regular, connected and pancyclic subgraphs. We prove that for n≥4 and 2n−1=n1+n2 with n1,n2≥2, AQn can be decomposed into two spanning subgraphs H1 and H2 such that Hi is ni-regular and ni-connected for i=1,2. Moreover, Hi is 4-pancyclic if ni≥3.

The conditional $h$-vertex($h$-edge) connectivity of a connected graph $H$ of minimum degree $ k > h$ is the size of a smallest vertex(edge) set $F$ of $H$ such that $H - F$ is a disconnected graph of minimum degree at least $h.$ Let $G$ be the Cartesian product of $r\geq 1$ cycles, each of length at least four and let $h$ be an integer such that $...

The conditional h-vertex (h-edge) connectivity of a connected graph H of minimum degree k > h is the size of a smallest vertex (edge) set F of H such that H − F is a disconnected graph of minimum degree at least h. Let G be the Cartesian product of r ≥ 1 cycles, each of length at least four and let h be an integer such that 0 ≤ h ≤ 2r − 2. In this...

Zaslavsky introduced the concept of lifted-graphic matroid. For binary matroids, a binary elementary lift can be defined in terms of the splitting operation. In this paper, we give a method to get a forbidden-minor characterization for the class of graphic matroids whose all lifted-graphic matroids are also graphic using the splitting operation.

It is known that the r-dimensional hypercube Q r can be decomposed into r-cycles and into 2r-cycles when r is even. We generalize these results to the class of the Cartesian product of cycles. We also prove that the k-ary r-cube Q k r , which is the Cartesian product of r k-cycles, can be decomposed into (tkr/2)-cycles if t divides k and 4 divides...

Given an $n$-connected binary matroid, we obtain a necessary and sufficient condition for its single-element coextensions to be $n$-connected.

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The $\Gamma$-extension operation on binary matroids is a generalization of the point-addition operation. In this paper, we obtain necessary and sufficient conditions to preserve $k$-connectedness of a binary matroid under the $\Gamma$-extension operation. We al...

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation. In general, under the Γ-extension operation the properties like graphicness and cographicness of matroids are not preserved. In this paper, we obtain forbidden...

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation. In this paper, we obtain necessary and sufficient conditions to preserve k-connectedness of a binary matroid under the Γ-extension operation. We also obtain a nece...

In this paper, we consider the problem of decomposing the augmented cube $AQ_n$ into two spanning, regular, connected and pancyclic subgraphs. We prove that for $ n \geq 4$ and $ 2n - 1 = n_1 + n_2 $ with $ n_1, n_2 \geq 2,$ the augmented cube $ AQ_n$ can be decomposed into two spanning subgraphs $ H_1$ and $ H_2$ such that each $ H_i$ is $n_i$-reg...

Let G be a graph obtained by taking the Cartesian product of finitely many cycles. It is known that G is bipancyclic, that is, G contains cycles of every even length from 4 to |V(G)|. We extend this result for the existence of 3-regular subgraphs in G. We prove that G contains a 3-regular, 2-connected subgraph with l vertices if l=8 or l=12 or l is...

An H-factorization of a graph G is a partition of the edge set of G into spanning subgraphs (or factors) each of whose components are isomorphic to a graph H. Let G be the Cartesian product of the cycles C1,C2,…,Cn with |Ci|=2ki≥4 for each i. El-Zanati and Eynden proved that G has a C-factorization, where C is a cycle of length s, if and only if s=...

It is known that the $n$-dimensional hypercube $Q_n,$ for $n$ even, has a decomposition into $k$-cycles for $k=n, 2n,$ $2^l$ with $2 \leq l \leq n.$ In this paper, we prove that $Q_n$ has a decomposition into $2^mn$-cycles for $n \geq 2^m.$ As an immediate consequence of this result, we get path decompositions of $Q_n$ as well. This gives a partial...

In general, the splitting operation on a binary matroid M does not preserve the connectivity of M. In this paper, we provide sufficient conditions to preserve n-connectedness of a binary matroid under splitting operation. As a consequence, for an (n+1)-connected binary matroid M, we give a precise characterization of when the splitting matroid MT i...

The Cartesian product of n cycles is a 2n-regular, 2n-connected and bipancyclic graph. Let G be the Cartesian product of n even cycles and let 2n = n1 + n2 + · · · + nk with k ≥ 2 and ni ≥ 2 for each i. We prove that if k = 2, then G can be decomposed into two spanning subgraphs G1 and G2 such that each Gi is ni-regular, ni-connected, and bipancycl...

We consider the problem of determining the possible orders for k-regular, k-connected and bipancyclic subgraphs of the hypercube Qn. For k = 2 and k = 3, the solution to the problem is known. In this paper, we solve the problem for k = 4 by proving that Qn has a 4-regular, 4-connected and bipancyclic subgraph on l vertices if and only if l = 16 or...

It is known that the [Formula: see text]-dimensional hypercube [Formula: see text] for [Formula: see text] with [Formula: see text] can be decomposed into two spanning bipancyclic subgraphs [Formula: see text] and [Formula: see text] such that [Formula: see text] is [Formula: see text]-regular and [Formula: see text]-connected for [Formula: see tex...

An m-factorization of a graph is a decomposition of its edge set into edge-disjoint m-regular spanning subgraphs (or factors). In this paper, we prove the existence of an isomorphic m-factorization of the Cartesian product of graphs each of which is decomposable into Hamiltonian even cycles. Moreover, each factor in the m-factorization is m-connect...

The splitting-off operation has important applications for graph connectivity problems. Shikare, Dalvi, and Dhotre [splitting-off operation for binary matroids and its applications, Graphs and Combinatorics, 27(6) (2011), 871-882] extended this operation to binary matroids. In this paper, we provide a sufficient condition for preserving n-connected...

In this paper, we consider the problem of decomposing the edge set of the hypercube Qn into two spanning, regular, connected, bipancyclic subgraphs. We prove that if n = n1 + n2 with n1 ≥ 2 and n2 ≥ 2, then the edge set of Qn can be decomposed into two spanning, bipancyclic subgraphs H1 and H2 such that Hi is ni-regular and ni-connected for i = 1,...

The -dimensional hypercube is bipancyclic; that is, it contains a cycle of every even length from 4 to . In this paper, we prove that contains a 3-regular, 3-connected, bipancyclic subgraph with vertices for every even from 8 to except 10.

Let M be a simple connected binary matroid with corank at least two such that M has no connected hyperplane. Seymour proved that M has a non-trivial series class. We improve this result by proving that M has at least two disjoint non-trivial series classes L-1 and L-2 such that both M\L-1 and M\L-2 are connected. Our result extends the correspondin...

In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the splitting operation, by every pair of elements, on M yields a cographic matroid. This problem is solved by proving that there are exactly three minor-minimal graphs that do not have this property.

This paper is based on the splitting operation for binary matroids that was introduced by Raghunathan, Shikare, and Waphare [Discrete Math. 184 (1998), p. 267-271] as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which cographic matroids M have the property that th...

We provide a sufficient condition for the existence of a cycle in a connected graph which is edge-disjoint from two connected subgraphs and of such that is connected.

J. G. Conlon [J. Graph Theory 45, No. 3, 163–223 (2004; Zbl 1033.05062)] proved that there exists an even cycle C in a 3-connected graph G≇K 5 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 C-E(C) is 2-connected and...

Using a splitting operation and a splitting lemma for connected graphs, Fleischner characterized connected Eulerian graphs. In this paper, we obtain a splitting lemma for 2-connected graphs and characterize 2-connected Eulerian graphs. As a consequence, we characterize connected graphic Eulerian matroids.

We prove that if M is a connected and vertically 3-connected binary matroid of cogirth at least 4 and girth at least 3, then the matroid obtained by splitting away any pair of elements of M is connected.

We prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M(K 4 ).

We obtain a forbidden-minor characterization of the class of binary gammoids M such that, for every pair of elements x,y of M, both the splitting matroid M x,y and the element splitting matroid M x,y ' are binary gammoids.

The element set splitting operation for binary matroids has been introduced by Azanchiler [2] as a natural generalization of the corresponding operation in graphs. In this paper, we explore the effect of this operation on graphic and cographic matroids. Key words and phrasesBinary matroid-graphic matroid-minor-element splitting operation-es-splitt...

Kriesell proved that every almost critical graph of connectivity 2 nonisomorphic to a cycle has at least 2 removable ears of length greater than 2. We improve this lower bound on the number of removable ears. A necessary condition for critically 2-connected graphs in terms of a forbidden minor is obtained. Further, we investigate properties of a sp...

This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M...

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 ob...

A series class P of a connected matroid M is removable if M∖P is connected. In this paper, we prove that a connected matroid M with r * (M)≥2 has at least r * (M)+1 removable series classes. Further, we obtain certain results from which the following result of Oxley and its graph theoretic version follow: If C is a circuit of a connected matroid M...

A cycle C in a graph G is called non-separating if G-V(C) is connected, and called removable if G-E(C) is 2-connected. We characterize connected simple graphs of minimum degree at least 3 having two vertex disjoint non-separating cycles. Further, we prove that every 2-connected simple graph of minimum degree at least 4, except the complete graph K...