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The vertex monophonic number of a graph

100%

Santhakumaran A., Titus P.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 2

191-204

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x -y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mₓ(G). An x-monophonic set of cardinality mₓ(G) is called a mₓ-set of G. We determine bounds for it and characterize graphs which realize these bounds. A connected graph of order p with vertex monophonic numbers either p - 1 or p - 2 for every vertex is characterized. It is shown that for positive integers a, b and n ≥ 2 with 2 ≤ a ≤ b, there exists a connected graph G with radₘG = a, diamₘG = b and mₓ(G) = n for some vertex x in G. Also, it is shown that for each triple m, n and p of integers with 1 ≤ n ≤ p -m -1 and m ≥ 3, there is a connected graph G of order p, monophonic diameter m and mₓ(G) = n for some vertex x of G.

The connected forcing connected vertex detour number of a graph

100%

Santhakumaran A., Titus P.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 3

461-473

For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdₓ(G). For a minimum connected x-detour set Sₓ of G, a subset T ⊆ Sₓ is called a connected x-forcing subset for Sₓ if the induced subgraph G[T] is connected and Sₓ is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sₓ of minimum cardinality is a minimum connected x-forcing subset of Sₓ. The connected forcing connected x-detour number of Sₓ, denoted by $cf_{cdx}(Sₓ)$, is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is $cf_{cdx}(G) = mincf_{cdx}(Sₓ)$, where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Santhakumaran A.

2 Titus P.

1 2012

1 2011

Wyniki wyszukiwania

The vertex monophonic number of a graph

The connected forcing connected vertex detour number of a graph