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E. G. K. M. Gamlath

Publications and source records attributed to E. G. K. M. Gamlath.

4 recordsLinked to original sources
arXiv2025

Independent Bondage Number in Graphs under Girth Constraints

Given a finite, simple graph $G$, the independent bondage number of $G$ is the minimum size of an edge set such that its deletion results in a graph with strictly larger independent domination number than that of $G$. While the bondage number of graphs under girth constraints has been studied, very few results have yet been established for the independent bondage number. In this study, we establish upper bounds on the independent bondage number of planar graphs under given girth constraints, extending results on the bondage number by Fischermann, Rautenbach, and Volkmann and on the structures of planar graphs by Borodin and Ivanova. In particular, we identify additional structures and establish bounds on the independent bondage number for planar graphs with $δ(G) \geq 2$ and $g(G)\geq 5$, $δ(G)\geq 3$ and $g(G)\geq 4$, and $δ(G) \geq 2$ and $g(G)\geq 10$.

math.CO↗
arXiv2024

Graph Configurations and Independent Bondage Numbers of Planar Graphs

The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a graph with a larger independent domination number than that of G. In this research, we enhance the existing upper bound on the independent bondage number of a planar graph with a minimum degree of at least three by identifying specific configurations within such planar graphs.

math.CO↗
arXiv2024

On Total Bondage Number of Graphs

In this paper, we explore the concept of total bondage in finite graphs without isolated vertices. A vertex set $D$ is considered a total dominating set if every vertex $v$ in the graph $G$ has a neighbor in $D$. The minimum cardinality of all total dominating sets in $G$ is denoted as $γ_t(G)$. A total bondage edge set $B$ is a subset of the edges of $G$ such that the removal of $B$ from $G$ does not create isolated vertices, and the total dominating number of the resulting graph $G-B$ is strictly greater than $γ_t(G)$. The total bondage number of $G$, denoted $b_t(G)$, is defined as the minimum cardinality of such total bondage edge sets. Our paper establishes upper bounds on $b_t(G)$ based on the maximum degree of a graph. Notably, for planar graphs with minimum degree $δ(G) \geq 3$, we prove $b_t(G) \leq Δ+ 8$ or $b_t(G) \leq 10$. Additionally, for a connected planar graph with $δ(G) \geq 3$ and $g(G) \geq 4$, we show that $b_t(G) \leq Δ+ 3$ if $G$ does not contain an edge with degree sum at most 7. We also improve some upper bounds of the total bondage number for trees, enhance existing lemmas, and find upper bounds for total bondage in specific graph classes.

math.CO↗