Ramesh Hariharasubramanian

Biswajit Das, Ramesh Hariharasubramanian

A graph $G(V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that for distinct letters $x,y\in V$, $x$ and $y$ alternate in $w$ if and only if they are adjacent in $G$. In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are cliques. In this paper, we obtain necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most $3$, and that permutation graphs are the only co-bipartite graphs with representation number $2$. We prove that the speed is $2^{O(n \log n)}$ and the entropy is $0$. This provides an asymptotic bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

Eshwar Srinivasan, Ramesh Hariharasubramanian

In this work, we investigate the algorithmic aspects of two natural extensions of hereditary classes: the edge-apex class and the edge-add class, recently introduced by Singh and Sivaraman. These are defined as the graph classes obtained by at most one edge deletion or one non-edge addition, respectively, from a hereditary class $\mathcal{G}$. Building on earlier results showing that both classes remain hereditary and admit finite forbidden induced subgraph characterizations whenever $\mathcal{G}$ does, we focus on the Weighted Maximum Clique Problem (WMCP) and the Weighted Maximum Independent Set Problem (WMISP). We first present algorithms for WMCP and WMISP on both the edge-apex and edge-add classes of hereditary graph classes. Extending this framework, we introduce the notion of the $\mathcal{G}$-edge distance of a graph $G$, denoted by $ξ_{\mathcal{G}}(G)$, which quantifies how far $G$ is from the class $\mathcal{G}$ in terms of the minimum number of edge deletions or non-edge additions needed to transform it into a member of $\mathcal{G}$. By parameterizing with respect to this distance, we show that both WMCP and WMISP can be solved in $O^*(2^k)$ time on graphs whose $\mathcal{G}$-edge distance is $k$, provided these problems admit polynomial-time algorithms within the class $\mathcal{G}$. This result extends earlier algorithmic characterizations of the single edge-apex and edge-add classes to the more general setting of $k$-edge-distant graphs. By combining our general results with known properties of transitive graphs, we show that WMCP and WMISP can be solved in $O^*(2^k)$ time for graphs with transitive-edge distance $k$.

Suchanda Roy, Ramesh Hariharasubramanian

A graph $G=(V,E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that two distinct letters $x,y\in V$ alternate in $w$ if and only if $xy \in E$. Word-representable graphs form a well-studied graph class with connections to graph orientations, combinatorics on words, and graph coloring. A near-triangulation is a planar graph in which every face except the outer face is a triangle. Several subclasses of near-triangulations have previously been investigated in the context of word-representability, including polyomino triangulations, triangulations of rectangular polyominoes with a single domino tile, $K_4$-free near-triangulations, face subdivisions of triangular grid graphs, triangulations of grid-covered cylinder graphs, and chordal near-triangulations. In this paper, we obtain a complete characterization of word-representable near-triangulations in terms of forbidden induced subgraphs. Our result unifies and extends the previously known characterizations for the above subclasses, while also correcting inaccuracies appearing in earlier works.

Eshwar Srinivasan, Ramesh Hariharasubramanian

A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if and only if $(x,y) \in E(G)$. Equivalently, a graph is word-representable if and only if it admits a semi-transitive orientation, that is, an acyclic orientation in which, for every directed path $v_0 \rightarrow v_1 \rightarrow \cdots \rightarrow v_m$ with $m \ge 2$, either there is no arc between $v_0$ and $v_m$, or, for all $1 \le i < j \le m$, there exists an arc from $v_i$ to $v_j$. In this work, we provide a comprehensive structural and algorithmic characterization of word-representable co-bipartite graphs, a class of graphs whose vertex set can be partitioned into two cliques. This work unifies graph-theoretic and matrix-theoretic perspectives. We first establish that a co-bipartite graph is a circle graph if and only if it is a permutation graph, thereby deriving a minimal forbidden induced subgraph characterization for co-bipartite circle graphs. The central contribution then connects semi-transitivity with the circularly compatible ones property of binary matrices. In addition to the structural characterization, the paper introduces a linear-time recognition algorithm for semi-transitive co-bipartite graphs, utilizing Safe's matrix recognition framework.

Suchanda Roy, Ramesh Hariharasubramanian

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\dots,x_k)$ with $1\le x_1<x_2<\cdots<x_k\le n$, where two vertices $(x_1,\dots,x_k)$ and $(y_1,\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\le i\le k-1$ or $y_{i+1}=x_i$ for all $1\le i\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd girth. We further observe that shift graphs arise naturally as induced subgraphs of simplified de Bruijn graphs. Although simplified de Bruijn graphs contain non-word-representable members in general, we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable. As a consequence, we obtain an explicit family of graphs exhibiting a contrast between line graph and line digraph constructions: there exists a family of word-representable graphs whose line graphs are not word-representable when the number of vertices is at least $5$, while their line digraphs are word-representable.