arXiv subjects

K. V. Krishna

Publications and source records attributed to K. V. Krishna.

At least 19 recordsLinked to original sources

arXiv2026

Word-Representation of Melon Graphs

The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through the representation number, whereas the corresponding concept is the permutation-representation number for comparability graphs. The graphs with the (permutation-)representation number at most two were characterized in the literature. While certain examples in the class of graphs with the (permutation-)representation number three are known, no characterization for these classes is available. In this work, we prove that the representation number of melon graphs is at most three. Further, we characterize the class of melon graphs restricted to comparability graphs and show that their permutation-representation number is also at most three. Moreover, this work characterizes the word-representable line graphs of melon graphs and establishes that their representation number is at most three.

math.CO↗

arXiv2025

Mutually Abelian-Bordered Binary Words

A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and there exists a word that is a nonempty proper suffix of $u$ and prefix of $v$. Recently, Gabric studied the number of mutually bordered pairs. In this work, we extend the concept of mutually bordered pairs to abelian setting, and determine the number of mutually abelian-bordered pairs of binary words using lattice paths. We also find the number of unbordered pairs in this context.

math.CO↗

arXiv2025

Line Graphs of Non-Word-Representable Graphs are Not Always Non-Word-Representable

A graph is said to be word-representable if there exists a word over its vertex set such that any two vertices are adjacent if and only if they alternate in the word. If no such word exists, the graph is non-word-representable. In the literature, there are examples of non-word-representable graphs whose line graphs are non-word-representable. However, it is an open problem to determine whether the line graph of a non-word-representable graph is always non-word-representable or not? In this work, we address the open problem by considering a class of non-word-representable graphs, viz., Mycielski graphs of odd cycles of length at least five, and show that their line graphs are word-representable.

math.CO↗

arXiv2025

On the Conjecture of the Representation Number of Bipartite Graphs

While the problem of determining the representation number of an arbitrary word-representable graph is NP-hard, this problem is open even for bipartite graphs. The representation numbers are known for certain bipartite graphs including all the graphs with at most nine vertices. For bipartite graphs with partite sets of sizes $m$ and $n$, Glen et al. conjectured that the representation number is at most $\lceil \frac{m+n}{4}\rceil$, where $m+n \ge 9$. In this paper, we show that every bipartite graph is $\left( 1+ \lceil \frac{m}{2} \rceil \right)$-representable, where $m$ is the size of its smallest partite set. Furthermore, if $m$ is odd then we prove that the bipartite graphs are $\lceil \frac{m}{2} \rceil $-representable. Accordingly, we establish that the conjecture by Glen et al. holds good for all bipartite graphs leaving the bipartite graphs whose partite sets are of equal and even size. In case of the bipartite graphs with partite sets of equal and even size, we prove the conjecture for certain subclasses using the neighborhood inclusion graph approach.

math.CO↗

arXiv2025

Representation Number of Word-Representable Split Graphs

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. The word-representability of split graphs was studied in a series of papers in the literature, and the class of word-representable split graphs was characterized through semi-transitive orientation. Nonetheless, the representation number of this class of graphs is still not known. In general, determining the representation number of a word-representable graph is an NP-complete problem. In this work, through an algorithmic procedure, we show that the representation number of the class of word-representable split graphs is at most three. Further, we characterize the class of word-representable split graphs as well as the class of split comparability graphs which have representation number exactly three.

math.CO↗

arXiv2025

Characterization of Split Comparability Graphs

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A split comparability graph is a split graph which is transitively orientable. In this work, we characterize split comparability graphs in terms of vertex labelling. Further, using this characterization, we prove that the permutation-representation number of a split comparability graph is at most three. This gives us an alternative proof of the result in order theory that the dimension of a split order is at most three.

math.CO↗

arXiv2025

Word-Representability of Well-Partitioned Chordal Graphs

In this paper, we study the word-representability of well-partitioned chordal graphs using split decomposition. We show that every component of the minimal split decomposition of a well-partitioned chordal graph is a split graph. Thus we have a characterization for word-representability of well-partitioned chordal graphs. As a consequence, we prove that the recognition of word-representability of well-partitioned chordal graphs can be done in polynomial time. Moreover, we prove that the representation number of a word-representable well-partitioned chordal graph is at most three. Further, we obtain a minimal forbidden induced subgraph characterization of circle graphs restricted to well-partitioned chordal graphs. Accordingly, we determine the class of word-representable well-partitioned chordal graphs having representation number exactly three.

math.CO↗

arXiv2025

A Class of Non-Contracting Branch Groups with Non-Torsion Rigid Kernels

In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are very strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.

math.GR↗

arXiv2024

Characterization of Double-Arborescences and their Minimum-Word-Representants

A double-arborescence is a treelike comparability graph with an all-adjacent vertex. In this paper, we first give a forbidden induced subgraph characterization of double-arborescences, where we prove that double-arborescences are precisely $P_4$-free treelike comparability graphs. Then, we characterize a more general class consisting of $P_4$-free distance-hereditary graphs using split-decomposition trees. Consequently, using split-decomposition trees, we characterize double-arborescences and one of its subclasses, viz., arborescences; a double-arborescence is an arborescence if its all-adjacent vertex is a source or a sink. In the context of word-representable graphs, it is an open problem to find the classes of word-representable graphs whose minimum-word-representants are of length $2n - k$, where $n$ is the number of vertices of the graph and $k$ is its clique number. Contributing to the open problem, we devise an algorithmic procedure and show that the class of double-arborescences is one such class. It seems the class of double-arborescences is the first example satisfying the criteria given in the open problem, for an arbitrary $k$.

math.CO↗

arXiv2024

Characterization of Word-Representable Graphs using Modular Decomposition

In this work, we characterize the class of word-representable graphs with respect to the modular decomposition. Consequently, we determine the representation number of a word-representable graph in terms of the permutation-representation numbers of the modules and the representation number of the associated quotient graph. In this connection, we also obtain a complete answer to the open problem posed by Kitaev and Lozin on the word-representability of the lexicographical product of graphs.

math.CO↗

arXiv2024

Word-Representability of Graphs with respect to Split Recomposition

In this work, we show that the class of word-representable graphs is closed under split recomposition and determine the representation number of the graph obtained by recomposing two word-representable graphs. Accordingly, we show that the class of parity graphs is word-representable. Further, we obtain a characteristic property by which the recomposition of comparability graphs is a comparability graph. Consequently, we also establish the permutation-representation number (prn) of the resulting comparability graph. We also introduce a subclass of comparability graphs, called prn-irreducible graphs. We provide a criterion such that the split recomposition of two prn-irreducible graphs is a comparability graph and determine the prn of the resultant graph.

cs.DM↗

arXiv2023

On the Permutation-Representation Number of Bipartite Graphs using Neighborhood Graphs

The problems of determining the permutation-representation number (prn) and the representation number of bipartite graphs are open in the literature. Moreover, the decision problem corresponding to the determination of the prn of a bipartite graph is NP-complete. However, these numbers were established for certain subclasses of bipartite graphs, e.g., for crown graphs. Further, it was conjectured that the crown graphs have the highest representation number among the bipartite graphs. In this work, first, we reconcile the relation between the prn of a comparability graph and the dimension of its induced poset and review the upper bounds on the prn of bipartite graphs. Then, we study the prn of bipartite graphs using the notion called neighborhood graphs. This approach substantiates the aforesaid conjecture and gives us theoretical evidence. In this connection, we devise a polynomial-time procedure to construct a word that represents a given bipartite graph permutationally. Accordingly, we provide a better upper bound for the prn of bipartite graphs. Further, we construct a class of bipartite graphs, viz., extended crown graphs, defined over posets and investigate its prn using the neighborhood graphs.

cs.DM↗

arXiv2023

A Branch Group in a Class of Non-Contracting Weakly Regular Branch Groups

We provide a class of non-contracting groups containing an infinite family of fractal and weakly regular branch groups, and study certain properties including abelianization, just infiniteness, and word problem. We present an example of a branch group in this class and show that it is of exponential growth. It seems this is the first example of a non-contracting branch group constructed explicitly.

math.GR↗

arXiv2023

Words for the Graphs with Permutation-Representation Number at most Three

The graphs with permutation-representation number (\textit{prn}) at most two are known. While a characterization for the class of graphs with the \textit{prn} at most three is an open problem, we summarize the graphs of this class that are known so far. Although it is known that the \textit{prn} of trees is at most three, in this work, we devise a polynomial-time algorithm for obtaining a word representing a given tree permutationally. Consequently, we determine the words representing even cycles. Contributing to the class of graphs with the \textit{prn} at most three, we determine the \textit{prn} as well as the representation number of book graphs.

cs.DM↗

arXiv2021

On the Representation Number of Bipartite Graphs

A word-representable graph is a simple graph $G$ which can be represented by a word $w$ over the vertices of $G$ such that any two vertices are adjacent in $G$ if and only if they alternate in $w$. It is known that the class of comparability graphs -- the graphs which admit a transitive orientation -- is precisely the class of graphs that can be represented by a concatenation of permutations of vertices. The class of bipartite graphs is a subclass of comparability graphs. While it is an open problem to determine the representation number of comparability graphs, it was conjectured that the representation number of bipartite graphs on $n$ vertices is at most $n/4$. In this paper, we propose a polynomial time relabeling algorithm to produce a word representing a given bipartite graph which is a concatenation of permutations of the graph's vertices. Thus we obtain an upper bound for the representation number of bipartite graphs, which in turn gives us an upper bound for the dimension of the posets corresponding to bipartite graphs.

cs.DM↗

arXiv2018

On the structure of C-algebras through atomicity and if-then-else

This paper introduces the notions of atoms and atomicity in $C$-algebras and obtains a characterisation of atoms in the $C$-algebra of transformations. Further, this work presents some necessary conditions and sufficient conditions for the atomicity of $C$-algebras and shows that the class of finite atomic $C$-algebras is precisely that of finite adas. This paper also uses the if-then-else action to study the structure of $C$-algebras and classify the elements of the $C$-algebra of transformations.

cs.LO↗

arXiv2017

On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian $p$-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently the minimum disconnecting sets of power graphs of the aforementioned groups are determined. Then the equality of connectivity and minimum degree of power graphs of finite groups is investigated and in this connection, certain necessary conditions are produced. A necessary and sufficient condition for the equality of connectivity and minimum degree of power graphs of finite cyclic groups is obtained. Moreover, the equality is examined for the power graphs of abelian $p$-groups, dihedral groups and dicyclic groups.

math.CO↗

arXiv2017

On connectedness of power graphs of finite groups

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of $p$-groups are studied. In particular, the number of components of that of abelian $p$-groups are determined.

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