Identifying vertex covers in graphs

Michael A. Henning, Anders Yeo

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

Resumé

An identifying vertex cover in a graph G is a subset T of vertices in G that has a nonempty intersection with every edge of G such that T distinguishes the edges, that is, e∩T ≠ 0 for every edge e in G and e∩T ≠ f∩T for every two distinct edges e and f in G. The identifying vertex cover number TD(G) of G is the minimum size of an identifying vertex cover in G. We observe that TD(G)+ρ(G) = |V (G)|, where ρ(G) denotes the packing number of G. We conjecture that if G is a graph of order n and size m with maximum degree Δ, then TD(G) ≤(Δ(Δ-1)/ Δ2+1)n + (2/Δ2+1) m. If the conjecture is true, then the bound is best possible for all Δ ≥ 1. We prove this conjecture when Δ ≥ 1 and G is a Δ-regular graph. The three known Moore graphs of diameter 2, namely the 5-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when Δ∈ {2; 3}.

OriginalsprogEngelsk
Artikelnummer32
TidsskriftElectronic Journal of Combinatorics
Vol/bind19
Udgave nummer4
Antal sider13
ISSN1097-1440
StatusUdgivet - 2012
Udgivet eksterntJa

Fingeraftryk

Vertex Cover
Graph in graph theory
Regular Graph
Petersen Graph
Maximum Degree
Packing
Equality
Intersection
Upper bound
Denote
Distinct
Cycle
Subset

Citer dette

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Identifying vertex covers in graphs. / Henning, Michael A.; Yeo, Anders.

I: Electronic Journal of Combinatorics, Bind 19, Nr. 4, 32, 2012.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Identifying vertex covers in graphs

AU - Henning, Michael A.

AU - Yeo, Anders

PY - 2012

Y1 - 2012

N2 - An identifying vertex cover in a graph G is a subset T of vertices in G that has a nonempty intersection with every edge of G such that T distinguishes the edges, that is, e∩T ≠ 0 for every edge e in G and e∩T ≠ f∩T for every two distinct edges e and f in G. The identifying vertex cover number TD(G) of G is the minimum size of an identifying vertex cover in G. We observe that TD(G)+ρ(G) = |V (G)|, where ρ(G) denotes the packing number of G. We conjecture that if G is a graph of order n and size m with maximum degree Δ, then TD(G) ≤(Δ(Δ-1)/ Δ2+1)n + (2/Δ2+1) m. If the conjecture is true, then the bound is best possible for all Δ ≥ 1. We prove this conjecture when Δ ≥ 1 and G is a Δ-regular graph. The three known Moore graphs of diameter 2, namely the 5-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when Δ∈ {2; 3}.

AB - An identifying vertex cover in a graph G is a subset T of vertices in G that has a nonempty intersection with every edge of G such that T distinguishes the edges, that is, e∩T ≠ 0 for every edge e in G and e∩T ≠ f∩T for every two distinct edges e and f in G. The identifying vertex cover number TD(G) of G is the minimum size of an identifying vertex cover in G. We observe that TD(G)+ρ(G) = |V (G)|, where ρ(G) denotes the packing number of G. We conjecture that if G is a graph of order n and size m with maximum degree Δ, then TD(G) ≤(Δ(Δ-1)/ Δ2+1)n + (2/Δ2+1) m. If the conjecture is true, then the bound is best possible for all Δ ≥ 1. We prove this conjecture when Δ ≥ 1 and G is a Δ-regular graph. The three known Moore graphs of diameter 2, namely the 5-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when Δ∈ {2; 3}.

KW - Identifying vertex cover

KW - Transversal

KW - Vertex cover

M3 - Journal article

VL - 19

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1097-1440

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M1 - 32

ER -