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Open AccessArticle

Exploring Geometrical Properties of Annihilator Intersection Graph of Commutative Rings

by Ali Al Khabyah and Moin A. Ansari

Axioms 2025, 14(5), 336; https://doi.org/10.3390/axioms14050336 - 27 Apr 2025

Cited by 1 | Viewed by 481

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not [...] Read more.

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not directed in nature, where the vertex set is represented by

D {(Λ)}^{*}

. There is a connection in the form of an edge between two distinct vertices

ς

and

ϱ

AIG (Λ)

iff

A n n (ς ϱ) \neq A n n (ς) \cap A n n (ϱ)

. In this work, we begin by categorizing commutative rings

Λ

, which are finite in structure, so that

AIG (Λ)

forms a star graph/2-outerplanar graph, and we identify the inner vertex number of

AIG (Λ)

. In addition, a classification of the finite rings where the genus of

AIG (Λ)

is 2, meaning

AIG (Λ)

is a double-toroidal graph, is also investigated. Further, we determine

Λ

, having a crosscap 1 of

AIG (Λ),

indicating that

AIG (Λ)

is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two. Full article

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12 pages, 307 KB

Open AccessArticle

On Two Open Problems on Double Vertex-Edge Domination in Graphs

by Fang Miao, Wenjie Fan, Mustapha Chellali, Rana Khoeilar, Seyed Mahmoud Sheikholeslami and Marzieh Soroudi

Mathematics 2019, 7(11), 1010; https://doi.org/10.3390/math7111010 - 24 Oct 2019

Cited by 1 | Viewed by 2338

Abstract

A vertex v of a graph

G = (V, E)

, ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set

S \subseteq V

is a double vertex-edge dominating set if [...] Read more.

A vertex v of a graph

G = (V, E)

, ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set

S \subseteq V

is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number

γ_{d v e} (G)

is the minimum cardinality of a double vertex-edge dominating set in G. A subset

S \subseteq V

is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in

V - S

has at least two neighbors in S). The total domination number

γ_{t} (G)

is the minimum cardinality of a total dominating set of G, and the 2-domination number

γ_{2} (G)

is the minimum cardinality of a 2-dominating set of

G .

Krishnakumari et al. (2017) showed that for every triangle-free graph

G,

γ_{d v e} (G) \leq γ_{2} (G)

, and in addition, if G has no isolated vertices, then

γ_{d v e} (G) \leq γ_{t} (G) .

Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with

γ_{d v e} (T) = γ_{t} (T)

γ_{d v e} (T) = γ_{2} (T)

. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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