arXiv subjects

Nishad Kothari

Publications and source records attributed to Nishad Kothari.

14 recordsLinked to original sources

arXiv2026

Cubic graphs, $S$-minors and conformal minors

It is well-known that any class of simple graphs, that is characterized by finitely many forbidden minors, also admits a characterization by finitely many forbidden topological minors; furthermore, the list of forbidden topological minors may be derived from the list of forbidden minors. We prove a similar result in Matching Theory. Our Main Theorem states that any class of matching covered graphs, that is characterized by finitely many forbidden $S$-minors that are cubic, also admits a characterization by finitely many forbidden conformal minors that are cubic as well; once again, the list of forbidden conformal minors may be derived from the list of forbidden $S$-minors. In order to establish the above, we first prove that every matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the $\Theta$ graph (that is, two vertices joined by three edges). (In fact, we need and prove a much stronger statement.) This is reminiscent of a theorem due to Lov\'asz: every nonbipartite matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the triangular prism $\overline{C_6}$. As applications of our Main Theorem, we deduce known 'forbidden conformal minor characterizations' of pfaffian near-bipartite graphs, and of pfaffian solid graphs, using their respective known 'forbidden $S$-minor characterizations'.

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arXiv2025

$\lambda$-matchability in cubic graphs

A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices. Clearly, $\lambda=0$ for bipartite graphs; ergo, we define $\lambda$-matchable pairs analogously, and we let $\rho(G)$ denote the number of such pairs. We improve the constant lower bounds on both $\lambda$ and $\rho$ established recently by Chen, Lu and Zhang [Discrete Math., 2025] using matching-theoretic parameters arising from the seminal work of Lov\'asz [J. Combin. Theory Ser. B, 1987], and we characterize all of the tight examples. We also solve the problem posed by Chen, Lu and Zhang: characterize 2-connected cubic graphs each of whose vertices is $\lambda$-matchable.

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arXiv2024

Solitude patterns of $r$-graphs via CLM's dependence relation

Recently, Goedgebeur, Mazzuoccolo, Renders, Wolf and the second author [Cubic graphs with edges in exactly one perfect matching, J. Graph Theory 112 (2026), 276-289] proved that every $3$-connected $3$-regular graph has at most six solitary edges; an edge is solitary if it participates in precisely one perfect matching. They also gave complete characterizations of $3$-connected $3$-regular graphs that have $k$ solitary edges for each $k \ge 3$. A connected $r$-regular graph, where $r \geq 3$, is an $r$-graph if each odd cut has at least $r$ edges; for instance, $3$-graphs are precisely the $2$-connected $3$-regular graphs. We generalize the bound and characterizations, mentioned in the preceding paragraph, to $3$-edge-connected $r$-graphs of order four or more; in particular, for $r \geq 4$, we establish stronger bounds:\ (i) at most four solitary edges, and (ii) if the graph is simple then it is devoid of solitary edges. Apart from characterizing all $3$-edge-connected $r$-graphs that have three or more solitary edges, for the case $r=3$, we provide a recursive characterization of those that have precisely two solitary edges such that they lie in the same perfect matching. Finally, for all $r$-graphs (not necessarily $3$-edge-connected), we establish that: (i) every such graph, except for a few small graphs, has at most $\frac{n}{2}$ solitary edges, and (ii) every such graph decomposes uniquely into $3$-edge-connected $r$-graphs, and its solitary edges may be computed recursively via this decomposition. Our proofs and insights rely heavily on the dependence and mutual dependence relationships introduced and investigated by Carvalho, Lucchesi and Murty [Ear decompositions of matching covered graphs, Combinatorica 19 (1999), 151-174].

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arXiv2024

$\theta$-free matching covered graphs: characterization and consequences

The Ear Decomposition Theorem of Lov\'asz & Plummer (1986) implies that every matching covered graph (MCG), except $K_2$ and cycles, contains (at least) one of $\theta$ and $K_4$ as a conformal minor. Lov\'asz [Combinatorica 1983] proved the refinement that every nonbipartite MCG contains one of $K_4$ and $\overline{C_6}$. These immediately lead to three problems: characterize (i) $\theta$-free graphs, (ii) $K_4$-free graphs and (iii) $\overline{C_6}$-free graphs. Kothari and Murty [JGT 2016] used the tight cut decomposition theory to solve the planar case of (ii) and (iii); the nonplanar cases are open. In contrast, we exploit a seminal result of Edmonds, Lov\'asz and Pulleyblank [Combinatorica 1982] to obtain a structural characterization of $\theta$-free graphs that immediately places the corresponding decision problem in P. The Petersen graph plays a key role. We deduce that every $\theta$-free graph has at most $2n-2$ edges, and we characterize the tight examples. Despite being sparse, these graphs are not necessarily planar. In the style of Little [JCT-B 1975], we characterize Pfaffian $\theta$-free graphs in terms of their forbidden conformal minors. Using the works of Robertson, Seymour and Thomas [Ann. of Math. 1999], and of McCuaig [E-JC 2004], we deduce that the Pfaffian recognition problem is in P for $\theta$-free graphs. Deciding whether a cubic graph is 3-edge-colorable is NP-complete; for $\theta$-free ones, we provide a characterization of those that are 3-edge-colorable, and deduce that the corresponding decision problem lies in P. McCuaig [JGT 2000] characterized 3-connected bipartite cubic graphs each of whose conformal cycles is of length 2 $\pmod{4}$; the 2-connected case is open. We stumbled upon the serendipitous corollary of our main result that each conformal cycle of a 2-connected cubic graph is of length 0 $\pmod{4}$ if and only if it is $\theta$-free.

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arXiv2024

Planar cycle-extendable graphs

For most problems pertaining to perfect matchings, one may restrict attention to matching covered graphs - that is, connected nontrivial graphs with the property that each edge belongs to some perfect matching. There is extensive literature on these graphs that are also known as 1-extendable graphs (since each edge extends to a perfect matching) including an ear decomposition theorem due to Lov\'asz and Plummer. A cycle $C$ of a graph $G$ is conformal if $G-V(C)$ has a perfect matching; such cycles play an important role in the study of perfect matchings, especially when investigating the Pfaffian orientation problem. A matching covered graph $G$ is cycle-extendable if - for each even cycle $C$ - the cycle $C$ is conformal, or equivalently, each perfect matching of $C$ extends to a perfect matching of $G$, or equivalently, $C$ is the symmetric difference of two perfect matchings of $G$, or equivalently, $C$ extends to an ear decomposition of $G$. In the literature, these are also known as cycle-nice or as 1-cycle resonant graphs. Zhang, Wang, Yuan, Ng and Cheng, 2022, provided a characterization of claw-free cycle-extendable graphs. Guo and Zhang, 2004, and independently Zhang and Li, 2012, provided characterizations of bipartite planar cycle-extendable graphs. In this paper, we establish a characterization of all planar cycle-extendable graphs - in terms of $K_2$ and four infinite families.

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arXiv2024

Extremal minimal bipartite matching covered graphs

A connected graph, on four or more vertices, is matching covered (aka 1-extendable) if every edge is present in some perfect matching. An ear decomposition theorem exists for bipartite matching covered graphs due to Hetyei. From the results and proofs of Lov\'asz and Plummer, that rely on Hetyei's theorem, one may deduce that any minimal bipartite matching covered graph has at least $2(m-n+2)$ vertices of degree two (where minimal means that deleting any edge results in a graph that is not matching covered); such a graph is said to be extremal if it attains the stated lower bound. In this paper, we provide a complete characterization of the class of extremal minimal bipartite matching covered graphs. In particular, we prove that every such graph $G$ is obtained from two copies of a tree devoid of degree two vertices, say $T$ and $T'$, by adding edges -- each of which joins a leaf of $T$ with the corresponding leaf of $T'$. Apart from the aforementioned bound, there are four other bounds that appear in, or may be deduced from, the work of Lov\'asz and Plummer. Each of these bounds leads to a notion of extremality. In this paper, we obtain a complete characterization of all of these extremal classes and also establish relationships between them. Two of our characterizations are in the same spirit as the one stated above. For the remaining two extremal classes, we reduce each of them to one of the already characterized extremal classes using standard matching theoretic operations. A connected graph is k-extendable if it has a matching of cardinality $k$ and each such matching extends to a perfect matching. We also discuss bounds proved by Lou (1999) for minimal k-extendable bipartite graphs. We conjecture stronger bounds and provide evidence for our conjectures by constructing tight examples that are straightforward generalizations of the ones that appear in the 1-extendable case.

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arXiv2019

Equivalence classes in matching covered graphs

A connected graph $G$, of order two or more, is matching covered if each edge lies in some \pema. The tight cut decomposition of a matching covered graph $G$ yields a list of bricks and braces; as per a theorem of Lov{á}sz~\cite{lova87}, this list is unique (up to multiple edges); $b(G)$ denotes the number of bricks, and $c_4(G)$ denotes the number of braces that are isomorphic to the cycle $C_4$ (up to multiple edges). Two edges $e$ and $f$ are mutually dependent if, for each perfect matching $M$, $e \in M$ if and only if $f \in M$; Carvalho, Lucchesi and Murty investigated this notion in their landmark paper~\cite{clm99}. For any matching covered graph $G$, mutual dependence is an equivalence relation, and it partitions $E(G)$ into equivalence classes; this equivalence class partition is denoted by $\mathcal{E}_G$ and we refer to its parts as equivalence classes of $G$; we use $\varepsilon(G)$ to denote the cardinality of the largest equivalence class. The operation of `splicing' may be used to construct bigger matching covered graphs from smaller ones; see~\cite{lckm18}; `tight splicing' is a stronger version of `splicing'. (These are converses of the notions of `separating cut' and `tight cut'.) In this article, we answer the following basic question: if a matching covered graph $G$ is obtained by `splicing' (or by `tight splicing') two smaller matching covered graphs, say~$G_1$~and~$G_2$, then how is $\mathcal{E}_G$ related to $\mathcal{E}_{G_1}$ and to $\mathcal{E}_{G_2}$ (and vice versa)? As applications of our findings: firstly, we establish tight upper bounds on $\varepsilon(G)$ in terms of $b(G)$ and $c_4(G)$; secondly, we answer a recent question of He, Wei, Ye and Zhai~\cite{hwyz19}, in the affirmative, by constructing graphs that have arbitrarily high $κ(G)$~and~$\varepsilon(G)$ simultaneously, where $κ(G)$ denotes the vertex-connectivity.

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arXiv2019

Minimal Braces

McCuaig (2001, Brace Generation, J. Graph Theory 38: 124-169) proved a generation theorem for braces, and used it as the principal induction tool to obtain a structural characterization of Pfaffian braces (2004, P{\'o}lya's Permanent Problem, Electronic J. Combinatorics 11: R79). A brace is minimal if deleting any edge results in a graph that is not a brace. From McCuaig's brace generation theorem, we derive our main theorem that may be viewed as an induction tool for minimal braces. As an application, we prove that a minimal brace of order $2n$ has size at most $5n-10$, when $n \geq 6$, and we provide a complete characterization of minimal braces that meet this upper bound. A similar work has already been done in the context of minimal bricks by Norine and Thomas (2006, Minimal Bricks, J. Combin. Theory Ser. B 96: 505-513) wherein they deduce the main result from the brick generation theorem due to the same authors (2007, Generating Bricks, J. Combin. Theory Ser. B 97: 769-817).

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arXiv2018

Birkhoff-von Neumann Graphs that are PM-compact

A well-studied geometric object in combinatorial optimization is the perfect matching polytope of a graph $G$. In any investigation concerning the perfect matching polytope, one may assume that $G$ is matching covered --- that is, it is a connected graph (of order at least two) and each edge lies in some perfect matching. A graph $G$ is Birkhoff-von Neumann (BvN) if its perfect matching polytope is characterized solely by non-negativity and degree constraints. A result of Balas (1981) implies that $G$ is BvN if and only if $G$ does not contain a pair of vertex-disjoint odd cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. It follows immediately that the corresponding decision problem is in co-NP. However, it is not known to be in NP. The problem is in P if the input graph is planar --- due to a result of Carvalho, Lucchesi and Murty (2004). These authors, along with Kothari (2018), have shown that this problem is equivalent to the seemingly unrelated problem of deciding whether a given graph is $\overline{C_6}$-free. The combinatorial diameter of a polytope is the diameter of its $1$-skeleton graph. A graph $G$ is PM-compact (PMc) if the combinatorial diameter of its perfect matching polytope equals one. A result of Chv\'atal (1975) implies that $G$ is PMc if and only if $G$ does not contain a pair of vertex-disjoint even cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. Once again the corresponding decision problem is in co-NP, but it is not known to be in NP. The problem is in P if the input graph is bipartite or is near-bipartite --- due to a result of Wang, Lin, Carvalho, Lucchesi, Sanjith and Little (2013). In this paper, we consider the "intersection" of the aforementioned problems. We give a complete characterization of matching covered graphs that are BvN as well as PMc. (Thus the corresponding decision problem is in P.)

math.CO↗

arXiv2018

On essentially 4-edge-connected cubic bricks

Lov\'asz (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.) A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lov\'asz which states that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge. A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected and if its only $3$-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact. We prove that if $G$ is any essentially $4$-edge-connected cubic brick then its edge-set may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii) quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$ (distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.

math.CO↗

arXiv2017

On Two Unsolved Problems Concerning Matching Covered Graphs

A cut $C:=\partial(X)$ of a matching covered graph $G$ is a separating cut if both its $C$-contractions $G/X$ and $G/\overline{X}$ are also matching covered. A brick is solid if it is free of nontrivial separating cuts. In 2004, we (Carvalho, Lucchesi and Murty) showed that the perfect matching polytope of a brick may be described without recourse to odd set constraints if and only if it is solid. In 2006, we proved that the only simple planar solid bricks are the odd wheels. The problem of characterizing nonplanar solid bricks remains unsolved. A bi-subdivision of a graph $J$ is a graph obtained from $J$ by replacing each of its edges by paths of odd length. A matching covered graph $J$ is a conformal minor of a matching covered graph $G$ if there exists a bi-subdivision $H$ of $J$ which is a subgraph of $G$ such that $G-V(H)$ has a perfect matching. For a fixed matching covered graph $J$, a matching covered graph $G$ is $J$-based if $J$ is a conformal minor of $G$ and, otherwise, $G$ is $J$-free. A basic result due to Lov\'asz (1983) states that every nonbipartite matching covered graph is either $K_4$-based or is $\overline{C_6}$-based or both, where $\overline{C_6}$ is the triangular prism. In 2016, we (Kothari and Murty) showed that, for any cubic brick $J$, a matching covered graph $G$ is $J$-free if and only if each of its bricks is $J$-free. We also found characterizations of planar bricks which are $K_4$-free and those which are $\overline{C_6}$-free. Each of these problems remains unsolved in the nonplanar case. In this paper we show that the seemingly unrelated problems of characterizing nonplanar solid bricks and of characterizing nonplanar $\overline{C_6}$-free bricks are essentially the same. We do this by establishing that a simple nonplanar brick, other than the Petersen graph, is solid if and only if it is $\overline{C_6}$-free.

math.CO↗

arXiv2017

Generating simple near-bipartite bricks

A brick is a $3$-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick $G$ is near-bipartite if it has a pair of edges $\alpha$ and $\beta$ such that $G-\{\alpha,\beta\}$ is bipartite and matching covered; examples are $K_4$ and the triangular prism $\overline{C_6}$. The significance of near-bipartite bricks arises from the theory of ear decompositions of matching covered graphs. The object of this paper is to establish a generation procedure which is specific to the class of simple near-bipartite bricks. In particular, we prove that every simple near-bipartite brick $G$ has an edge $e$ so that the graph obtained from $G-e$ by contracting each edge that is incident with a vertex of degree two is also a simple near-bipartite brick, unless $G$ belongs to any of eight well-defined infinite families. This is a refinement of the brick generation theorem of Norine and Thomas (Generating Bricks, J. Combin. Theory Ser. B, 2007) which is appropriate for the restricted class of near-bipartite bricks. Earlier, the first author (Generating near-bipartite bricks, J. Graph Theory, 2019) proved a similar generation theorem for (not necessarily simple) near-bipartite bricks; we deduce our main result from this theorem. Our proof is based on the strategy of Carvalho, Lucchesi and Murty (2008) and uses several of their techniques and results. The results presented here also appear in the Ph.D. thesis of the first author.

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arXiv2016

Generating Near-Bipartite Bricks

A $3$-connected graph $G$ is a brick if, for any two vertices $u$ and $v$, the graph $G-\{u,v\}$ has a perfect matching. Deleting an edge $e$ from a brick $G$ results in a graph with zero, one or two vertices of degree two. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of $G-e$ is the graph $J$ obtained from it by bicontracting all its vertices of degree two. An edge $e$ is thin if $J$ is also a brick. Carvalho, Lucchesi and Murty [How to build a brick, Discrete Mathematics 306 (2006), 2383-2410] showed that every brick, distinct from $K_4$, the triangular prism $\overline{C_6}$ and the Petersen graph, has a thin edge. Their theorem yields a generation procedure for bricks, using which they showed that every simple planar solid brick is an odd wheel. A brick $G$ is near-bipartite if it has a pair of edges $\alpha$ and $\beta$ such that $G-\{\alpha,\beta\}$ is bipartite and matching covered; examples are $K_4$ and $\overline{C_6}$. The significance of near-bipartite graphs arises from the theory of ear decompositions of matching covered graphs. The object of this paper is to establish a generation procedure which is specific to the class of near-bipartite bricks. In particular, we prove that if $G$ is any near-bipartite brick, distinct from $K_4$ and $\overline{C_6}$, then $G$ has a thin edge $e$ so that the retract $J$ of $G-e$ is also near-bipartite. In a subsequent work, with Marcelo H. de Carvalho, we use the results of this paper to prove a generation theorem for simple near-bipartite bricks.

math.CO↗

arXiv2011

Approximation Algorithms for Digraph Width Parameters

Several problems that are NP-hard on general graphs are efficiently solvable on graphs with bounded treewidth. Efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs. Directed treewidth, DAG-width and Kelly-width are some such notions which generalize treewidth, whereas directed pathwidth generalizes pathwidth. Each of these digraph width measures have an associated decomposition structure. In this paper, we present approximation algorithms for all these digraph width parameters. In particular, we give an O(sqrt{logn})-approximation algorithm for directed treewidth, and an O({\log}^{3/2}{n})-approximation algorithm for directed pathwidth, DAG-width and Kelly-width. Our algorithms construct the corresponding decompositions whose widths are within the above mentioned approximation factors.

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