Detour Set Research Articles

Connected Restrained Detour Number of a Graph

Objectives: To find the connected restrained detour number for standard graphs and mesh graphs. Methods: By determining the connected restrained detour set with minimum cardinality, the connected restrained detour number of a graph is investigated. Findings: We study that the connected restrained detour number of the graphs is altered when we add pendent vertices. The minimum and maximum degree vertices of a graph are deleted and the connected restrained detour number of the mesh graph is computed. Novelty: Finding the detour path plays a vital role in the network-based systems. Planning the largest route that is connected and restrained is essential in business, industries and radio technologies. We introduce the new concept of connected restrained detour number. We also exhibit the bounds for the connected restrained detour set of a graph. Keywords: Detour Set; Detour Number; Mesh Graphs; Connected Restrained Detour Set; Connected Restrained Detour Number

Weak edge triangle free detour number of a graph

For any two vertices u and v in a connected graph G = ( V , E ), a u − v path P is called a u − v triangle free path if no three vertices of P induce a triangle. The triangle free detour distance D △ f ( u , v ) is the length of a longest u − v triangle free path in G . A u − v path of length D △ f ( u , v ) is called a u − v triangle free detour . A set S ⊆ V is called a weak edge triangle free detour set of G if every edge of G has both ends in S or it lies on a triangle free detour joining a pair of vertices of S . The weak edge triangle free detour number w d n △ f ( G ) of G is the minimum order of its weak edge triangle free detour sets and any weak edge triangle free detour set of order w d n △ f ( G ) is a weak edge triangle free detour basis of G . Certain properties of these concepts are studied. The weak edge triangle free detour numbers of certain classes of graphs are determined. Its relationship with the triangle free detour diameter is discussed and it is proved that for any three positive integers a , b and n of integers with 3 ≤ b ≤ n − a + 1 and a ≥ 4, there exists a connected graph G of order n with triangle free detour diameter D △ f = a and w d n △ f ( G )= b . It is also proved that for any three positive integers a , b and c with 3 ≤ a ≤ b and c ≥ b + 2, there exists a connected graph G such that R △ f = a , D △ f = b and w d n △ f ( G )= c .

The Upper Total Triangle Free Detour Number of a Graph

For a connected graph G = (V,E) of order at least two, a total triangle free detour set of a graph G is a triangle free detour set S such that the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total triangle free detour set of G is the total triangle free detour number of G. It is denoted by 〖tdn〗_Δf (G). A total triangle free detour set of cardinality 〖tdn〗_Δf (G) is called 〖tdn〗_Δf- set of G. In this article, the concept of upper total triangle free detour number of a graph G is introduced. It is found that the upper total triangle free detour number differs from total triangle free detour number. The upper total triangle free detour number is found for some standard graphs. Their bounds are determined. Certain general properties satisfied by them are studied.

Upper triangle free detour number of a graph

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text] Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

Graphs of edge-to-vertex detour number 2

For two vertices u and v in a graph G = (V,E), the detour distance D(u, v) is the length of a longest u − v path in G. A u − v path of length D(u, v) is called a u − v detour. For subsets A and B of V , the detour distance D(A,B) is defined as D(A,B) = min{D(x, y) : x ∈ A, y ∈ B}. A u − v path of length D(A,B) is called an A-B detour joining the sets A,B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A − B detour if x is a vertex of some A−B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn2(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn2(G) is an edge-to-vertex detour basis of G. Graphs G of size q for which dn2(G) = 2 are characterized.

The open detour number of a graph

A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text].

A removability theorem for Sobolev functions and detour sets

We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called "detour sets", which resemble the Sierpi\'nski gasket. The main theorem is that if $K\subset \mathbb R^n$ is a detour set and its complementary components are sufficiently regular, then $K$ is $W^{1,p}$-removable for $p>n$. Several examples and constructions of sets where the theorem applies are given, including the Sierpi\'nski gasket, Apollonian gaskets, and Julia sets.

The Connected Detour Numbers of Special Classes of Connected Graphs

Simple finite connected graphs G=V,E of p≥2 vertices are considered in this paper. A connected detour set of G is defined as a subset S⊆V such that the induced subgraph GS is connected and every vertex of G lies on a u−v detour for some u,v∈S. The connected detour number cdnG of a graph G is the minimum order of the connected detour sets of G. In this paper, we determined cdnG for three special classes of graphs G, namely, unicyclic graphs, bicyclic graphs, and cog-graphs for Cp, Kp, and Km,n.

Connected Weak Edge Detour Number of a Graph

Certain general properties of the detour distance, weak edge detour set, connected weak edge detour set, connected weak edge detour number and connected weak edge detour basis of graphs are studied in this paper. Their relationship with the detour diameter is discussed. It is proved that for each pair of integers k and n with 2 <= k <= n, there exists a connected graph G of order n with cdnw(G)=k. It is also proved that for any three positive integers R,D,k such that k >= D and R < D <= 2R, there exists a connected graph G with radD (G) = R, diamD G = D and cdnw(G)=k.

Upper Vertex Triangle Free Detour Number of a Graph

For a graph G, the x-triangle free detour set, the x-triangle free detour number, the minimal x-triangle free detour set, the upper x-triangle free detour number, are defined and studied. Certain bounds are determined and the relation with the vertex triangle free detour number of a graph is found out. Some realization problems, properties related to the upper vertex detour number, the upper vertex detour monophonic number and the upper vertex geodetic number are also studied.

The detour domination number of a graph

For a connected graph [Formula: see text], a set [Formula: see text] is called a detour dominating set of [Formula: see text], if [Formula: see text] is a detour set and dominating set of [Formula: see text]. The detour domination number [Formula: see text] of [Formula: see text] is the minimum order of its detour dominating sets and any detour dominating set of order [Formula: see text] is called a [Formula: see text] - set of [Formula: see text]. The detour domination numbers of some standard graphs are determined. Connected graph of order [Formula: see text] with detour domination number [Formula: see text] or [Formula: see text] is characterized. For positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph with [Formula: see text] and [Formula: see text].

Vertex Triangle Free Detour Number of a Graph

The \emph{$x$-triangle free detour number} $dn_{\triangle f_x}(G)$ of a connected graph $G$ is the minimum order of its $x$-triangle free detour sets and any $x$-triangle free detour set $S_{x} \subseteq V$ of order $dn_{\triangle f_x}(G)$ is a \emph{$x$-triangle free detour basis} of $G$. A connected graph of order $n$ with vertex triangle free detour number $n-1$ or $n-2$ for every vertex is characterized. Certain general properties satisfied by the vertex triangle free detour sets are studied.

The Total Detour Number of a Graph

For a connected graph G = (V, E), aset S ⊆ V(G) is called a total detour set of G if S is a detour set of G and the subgraph G[S] induced by S has no isolated vertex. The total detour number tdn(G) of G is the minimum order of its total detour sets and any total detour set of order tdn(G) is called a td-set of G. The total detour number of some standard graphs are determined. Necessary conditions for the total detour number to be 2 or P are given. It is shown that for any three integers a, b and p with 2 ≤ a < b ≤ p, there exists a connected graph G of order p with tdn (G) = a and cdn (G) = b. Also it is shown that for each triple D, k, p of integers with 3 ≤ k ≤ p - D + 1 and D ≥ 4, there exists a connected graph G of order P with detour diameter D and tdn (G) = k. A total detour set S in a connected graph G is called a minimal total detour set of G if no proper subset of S is a total detour set of G. The upper total detour number tdn+ (G) of G is the maximum cardinality of a minimal total detour set of G. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists a connected graph G with tdn (G) = a and tdn+ (G) = b.

The forcing connected detour number of a graph

For two vertices u and v in a graph G = (V, E), the detour distance D(u, v) is the length of a longest u—v path in G.A u—v path of length D(u, v) is called a u—v detour. A set ⊆ V is called a detour set of G if every vertex in G lies on a detour joining a pair of vertices of S.The detour number dn(G) of G is the minimum order of its detour sets and any detour set of order dn(G) is a detour basis of G.A set ⊆ V is called a connected detour set of G if S is detour set of G and the subgraph G[S] induced by S is connected. The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G.A subset T of a connected detour basis S is called a forcing subset for S if S is theuniquecon-nected detour basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G),is fcdn(G) = min{fcdn(S)},where the minimum is taken over all connected detour bases S in G. The forcing connected detour numbers ofcertain standard graphs are obtained. It is shown that for each pair a, b of integers with 0 ≤ a < b and b ≥ 3, there is a connected graph G with fcdn(G) = a and cdn(G) = b.