For a graph \(G=(V(G), E(G))\), a subset \(S \subset V(G)\) is a bipartite dominating set if every vertex in \(G-S\) is adjacent to a vertex in \(S\), and if the subgraph of \(G\) induced by \(S\) is bipartite. The bipartite domination number of \(G\), denoted by \(\gamma_{bip}(G)\), is the minimum cardinality of all bipartite dominating sets of \(G\). Xi and Yue [4] claimed that for every 2-connected outerplanar \(n\)-vertex graph \(G\), \(\gamma_{bip}(G) \leq \lceil \frac n 3 \rceil\), and that this bound is sharp. In this paper, correcting the result, we prove that \(\gamma_{bip}(G) \leq \lceil \frac 38 n \rceil\), where this bound is sharp.
Hartnell and Rall recently introduced the domatic number game. Alice and Bob color the vertices of a graph from a palette [\(k\)], and Alice wins if every color class is a dominating set at the end of the game. The largest winning palette size is denoted by \(\mathop{\mathrm{dom}}\nolimits_{g}(G)\) when Alice moves first and by \(\mathop{\mathrm{dom}}\nolimits'_{g}(G)\) when Bob moves first. Hartnell and Rall asked how these parameters behave under edge and vertex removal, and they also asked whether Bob can win with \(k\) colors while Alice wins with \(k+1\) colors. We give short answers. First, Alice-winning palettes are downward closed: if Alice can win with \(k+1\) colors, then she can win with \(k\) colors, in both versions of the game. Thus the proposed palette-size pathology never occurs. Second, if \(H\) is a spanning subgraph of \(G\), then \[ \mathop{\mathrm{dom}}\nolimits_{g}(H)\le \mathop{\mathrm{dom}}\nolimits_{g}(G),\qquad \mathop{\mathrm{dom}}\nolimits'_{g}(H)\le \mathop{\mathrm{dom}}\nolimits'_{g}(G). \] Thus edge deletion can never increase either invariant, and the inequalities may be strict. Finally, vertex deletion is not monotone: it can increase or decrease either invariant. Deleting one vertex can even increase either invariant by an arbitrarily large amount.
A graph \(G\) is said to be a an interval graph, if for each vertex \(u\) of \(G\), one can assign a set \(A_u\) which is a finite union of intervals on the real line such that \(u\) is adjacent to \(v\) in \(G\) if and only if \(A_u\cap A_v\neq\varnothing\). In this paper, we introduce a class of intersection graphs and show that it is equivalent to the class of interval graphs. We also investigate interval numbers of certain intersection graphs and establish several related results.
Motivated from the concept of strong regularity in the graph theory, few varieties of definitions for strongly regular signed graphs have been introduced. The initial one, which is due to Zaslavsky and the others are given by Stanic and Ramezani. The definition given by Stanic covers all the others. In this paper we provide some constructions for each of the definitions.
In this article, we obtain the determining number and the metric dimension of the zero-divisor graph of the ring of integers modulo \(n\) and of non-Boolean semisimple rings. For Boolean rings, an upper bound for these parameters is established. While the determining number and metric dimension of \(\Gamma(\mathbb{Z}_n)\) are known in the literature, we provide an alternative derivation based on a structural decomposition of the graph via generalized join. This approach offers a direct and unified method to compute these parameters. Further, we determine these parameters for joins of vertex-transitive graphs and investigate certain questions concerning the relationship between determining number and metric dimension.
