On the geodetic number of complementary prisms

Castonguay, Diane; Coelho, Erika M. M.; Coelho, Hebert; Nascimento, Julliano R.

doi:10.1016/j.ipl.2018.12.007

Cited by 12 publications

(7 citation statements)

References 21 publications

(17 reference statements)

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“…Haynes et al [17] investigated several graph theoretic properties of complementary prisms, such as independence, distance and domination. For further study on domination parameters in complementary prisms, see [7,8,14,19,20] and for certain other parameters see [1,6,28,34]. Cappelle et al [3] described a polynomial-time recognition algorithm for complementary prisms.…”

Section: Question: Does G Have An An Open-independent Open Locating D...mentioning

confidence: 99%

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Cappelle,

Coelho,

Foulds

et al. 2021

Preprint

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Let G = (V (G), E(G)) be a finite simple undirected graph with vertex set V (G), edge set E(G) and vertex subset S ⊆ V (G). S is termed open-dominating if every vertex of G has at least one neighbor in S, and open-independent, open-locating-dominating (an OLD oind -set for short) if no two vertices in G have the same set of neighbors in S, and each vertex in S is open-dominated exactly once by S. The problem of deciding whether or not G has an OLD oind -set has important applications, has been studied elsewhere and is known to be N P-complete. Reinforcing this result, it appears that the problem is notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs. Also, we present characterizations of both P 4 -tidy graphs and the complementary prisms of cographs, that have an OLD oind -set.

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Section: Question: Does G Have An An Open-independent Open Locating D...mentioning

confidence: 99%

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Cappelle,

Coelho,

Foulds

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In particular, the Petersen graph is the complementary prism

{C}_5\overline{C_5}

and

{K}_n\circ {K}_1

is the complementary prism

{K}_n\overline{K_n}

. From a structural geometrical point of view, complementary prism networks have been studied through their properties and indices, such that, chromatic index, 2 domination number, 3 cycle structure, 4 complexity properties, 5 spectral properties, 6 convexity number, 7 chromatic number, 8 etc.…”

Section: Introductionmentioning

confidence: 99%

Geometric and topological properties of the complementary prism networks

Méndez

Reyes

García

et al. 2023

Math Methods in App Sciences

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In this paper, we study the hyperbolicity constant of GG. In particular, we obtain upper and lower bounds for the hyperbolicity constant, and we compute its precise value for many graphs. Moreover, we obtain bounds and closed formulas for the general topological indices A(GG) = ∑ uv∈E(GG) a(d u , d v ) and B(GG) = ∑ u∈V(GG) b(d u ) (where d u denotes the degree of the vertex u, a is a symmetric function with real values, and b is a function with real values), and the generalized Wiener index W 𝜆 (GG), of complementary prisms networks. Finally, we performed a numerical study of the generalized Wiener index on random graphs.

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“…Haynes et al [17] investigated several graph theoretic properties of complementary prisms, such as independence, distance and domination. For further study on domination parameters in complementary prisms, see [14,7,8,19,20] and for certain other parameters see [1,6,28,35]. Cappelle et al [2] described a polynomial-time recognition algorithm for complementary prisms.…”

Section: Introductionmentioning

confidence: 99%

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Cappelle¹,

Coelho²,

Foulds³

et al. 2022

Discrete Mathematics &Amp; Theoretical Computer Science

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Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

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On the geodetic number of complementary prisms

Cited by 12 publications

References 21 publications

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Geometric and topological properties of the complementary prism networks

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

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