arXiv subjects

Shikhi M

Publications and source records attributed to Shikhi M.

6 recordsLinked to original sources

arXiv2026

k-Mutual Visibility in Graphs

In this paper, we introduce the notion of $k$-mutual visibility, a relaxation of classical mutual visibility in which every pair of selected vertices is joined by a shortest path containing at most $k$ internal vertices of the selected set. This parameterized concept naturally generalizes classical mutual visibility and provides a graded notion of obstruction tolerance. We define the $k$-mutual visibility number $\mu_k(G)$ and establish its fundamental properties. We derive bounds on $\mu_k(G)$ in terms of graph parameters such as diameter and girth, and determine its exact value for several fundamental graph classes. We further investigate $k$-mutual visibility in convex subgraphs and characterize it in block graphs by introducing the notion of $k$-admissible sets in the associated block-cutpoint tree. We present a polynomial-time algorithm, kMV, that recognizes whether a given subset $S\subseteq V(G)$ is a $k$-mutual visibility set of $G$. We also formulate the $k$-Mutual Visibility decision problem and prove that it is NP-complete. Finally, we define the $k$-mutual visibility covering number $\tau_k(G)$ and establish several of its fundamental properties.

math.CO↗

arXiv2026

Mutual-Visibility of Tree and Its Line Graphs

In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree $T\langle S\rangle$. For trees containing branch vertices, the notion of legs is introduced, and an explicit formula for the number of maximal mutual-visibility sets is derived in terms of the corresponding leg lengths. We prove that every tree is absolute-clear. It is further shown that, for every tree $T$ with at least two edges, the mutual-visibility number is preserved under the line graph operation, that is, $\mu(L(T))=\mu(T)$. Examples of unicyclic and block graphs for which this equality fails are also presented. Finally, a tight lower bound for the mutual-visibility number of the iterated line graph is established; namely, $\mu\bigl(L(L(T))\bigr)\ge \left\lfloor \frac{\Delta(T)^2}{3}\right\rfloor$.

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arXiv2025

Mutual visibility in Moore graphs and $(d,2)$-graphs with defect

The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we derive algebraic conditions for the mutual-visibility number of $(d,2)$-graphs with non-negative defect. We then determine this parameter for $(d,2,-2)$-graphs for $d=3$ and $4$, and establish an upper bound for $d=5$. In the case $\delta=0$, that is, for Moore graphs of diameter $2$, we focus on the Hoffman-Singleton graph. We establish an upper bound of $20$ for its mutual-visibility number and subsequently employ an integer programming approach to show that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman--Singleton graph is $10$.

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arXiv2025

Visibility Polynomial of Some Graph Classes

Mutual visibility in graphs provides a framework for analysing how vertices can observe one another along shortest paths free of internal obstructions. The visibility polynomial, which enumerates mutual-visibility sets of all orders, has emerged as a central invariant in this study, with both theoretical significance and practical relevance in areas such as surveillance, target tracking, and distributed coordination. While computing this polynomial is computationally demanding, with known complexity $O(n^32^n)$, explicit characterizations for specific graph families yield valuable structural insights. In this paper, we characterize the mutual-visibility sets of several fundamental graph classes and derive closed-form expressions for their associated visibility polynomials. These results deepen the understanding of visibility-based invariants and expand the toolkit available for studying visibility phenomena in networks.

math.CO↗

arXiv2025

Visibility polynomial of corona of two graphs

In multiagent systems, effective coordination, coverage, and communication often rely on the concept of visibility between agents or nodes within the system. Graph-theoretically, for any subset $X$ of vertices of a graph $G$, two vertices are said to be $X$-visible if there exists a shortest path between them that contains no vertex of $X$ as an internal vertex. In this paper, we investigate the visibility polynomial associated with the corona product of two graphs. The visibility polynomial encodes the number of mutual-visibility sets of all orders within a graph, and the process of enumerating these sets provides a deeper understanding of their structural properties. We characterize the structure of mutual-visibility sets arising specifically within the corona product. As part of this study, we introduce the notion of $C_Q$-visible sets, defined with respect to a selected subset $Q$ of vertices in a graph $G$. A $C_Q$-visible set is a collection of vertices in $\overline{Q}$ that is not only $Q$-visible, but also individually visible from each vertex in $Q$. Using this concept, we establish several characterizations and properties of mutual-visibility sets within the corona product, thereby providing deeper insights into their structure and behavior.

math.CO↗

arXiv2025

On the Visibility Polynomial of Graphs

Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if every pair of vertices in X are X-visible. The visibility polynomial of a graph G is defined as nu (G)=sum_{i >= 0} r_i x^i, where r_i denotes the number of mutual-visibility sets in G of cardinality i. In the present paper, the visibility polynomial is studied for some well-known classes of graphs. In particular, the instance at which the number of maximal mutual-visibility sets is equal for cycle graphs is identified. The visibility polynomial of the join of two graphs is studied. The algorithm for computing the visibility polynomial of a graph has been identified to have a time complexity of O(n^3.2^n) making the problem computationally intensive for larger graphs.

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