Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A . It is denoted by ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

Reconfiguring Dominating Sets in Minor-Closed Graph Classes

For a graph G, two dominating sets D and $$D'$$ D ′ in G, and a non-negative integer k, the set D is said to k-transform to $$D'$$ D ′ if there is a sequence $$D_0,\ldots ,D_\ell $$ D 0 , … , D ℓ of dominating sets in G such that $$D=D_0$$ D = D 0 , $$D'=D_\ell $$ D ′ = D ℓ , $$|D_i|\le k$$ | D i | ≤ k for every $$i\in \{ 0,1,\ldots ,\ell \}$$ i ∈ { 0 , 1 , … , ℓ } , and $$D_i$$ D i arises from $$D_{i-1}$$ D i - 1 by adding or removing one vertex for every $$i\in \{ 1,\ldots ,\ell \}$$ i ∈ { 1 , … , ℓ } . We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and $$D'$$ D ′ in G such that Dk-transforms to $$D'$$ D ′ only if $$k\ge \max \{ |D|,|D'|\}+c\sqrt{n}$$ k ≥ max { | D | , | D ′ | } + c n . Conversely, for every hereditary class $$\mathcal{G}$$ G that has balanced separators of order $$n\mapsto n^\alpha $$ n ↦ n α for some $$\alpha <1$$ α < 1 , we prove that there is some positive constant C such that, if G is a graph in $$\mathcal{G}$$ G of order n, and D and $$D'$$ D ′ are two dominating sets in G, then Dk-transforms to $$D'$$ D ′ for $$k=\max \{ |D|,|D'|\}+\lfloor Cn^\alpha \rfloor $$ k = max { | D | , | D ′ | } + ⌊ C n α ⌋ .

Grundy Domination of Forests and the Strong Product Conjecture

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a minimum partition of the non-isolate vertices of $F$ into caterpillars in which if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then such an edge must be incident to a non-leaf vertex in at least one of the caterpillars. We use this result to show the strong product conjecture of B. Brešar, Cs. Bujtás, T. Gologranc, S. Klavžar, G. Košmrlj, B.~Patkós, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like and toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests. Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph $G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that every non-complete connected graph contains a Grundy dominating set $S$ so that the induced subgraph of $S$ contains no isolated vertices.

Hamiltonicity of a class of toroidal graphs

If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.