Inapproximability results related to monophonic convexity

Toll number of the Cartesian and the lexicographic product of graphs

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by mc(G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance dm(u, v) from u to v is defined as the length of a longest u − v monophonic path in G. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius radmG of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diammG of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r <  d, there exists a connected graph G with radmG =  r, diammG =  d and mc(G) =  n. Also, if a,b and p are positive integers such that 2 ≤  a <  b ≤  p, then there exists a connected graph G of order p, m(G) =  a and mc(G) =  b.

The Connected Monophonic Number of a Graph

Let G be a connected graph of order at least two. We study about the monophonic sets and define a new set called monophonic radius-distance monophonic set of G. The monophonic radius-distance monophonic set of different kinds of graphs are found and verified. A new parameter monophonic radius-distance monophonic number is introduced and some bounds are found for several graphs. A relation between monophonic number and the monophonic radius-distance monophonic number of G is found. A main result, for every two integers c and d such that 2 ≤ c ≤ d, there is a graph with c as the monophonic number and d as the monophonic radius-distance monophonic number of G is proved.

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Monophonic number of a graph with respect to monophonic radius

Toll number of the strong product of graphs

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p 1 are characterized. For every pair a,b of positive integers with 2 � ab, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

On the monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x – y monophonic path is called an x – y detour monophonic path. A set S of vertices of G  is a detour monophonic set of G if each vertex v of G lies on an x - y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A total detour monophonic set of cardinality dmt(G) is called a dmt-set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dmt(G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dmt(G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p - 2, 3 ≤ k ≤ p and p – d – k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dmt(G) = k.

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The total detour monophonic number of a graph

For a connected graph $$G=(V, E)$$
 of order at least two, a total monophonic set of a graph $$G$$
 is a monophonic set $$S$$
 such that the subgraph $$G[S]$$
 induced by $$S$$
 has no isolated vertices. The minimum cardinality of a total monophonic set of $$G$$
 is the total monophonic number of $$G$$
 and is denoted by $${m}_{t}(G)$$
 . The number of extreme vertices and support vertices of $$G$$
 is its extreme-support order $$es(G)$$
 . A graph $$G$$
 is an extreme-support total monophonic graph if $${m}_{t}\left(G\right)=es(G)$$
 . Some interesting results on the extreme-support total monophonic graphs $$G$$
 are studied. Graphs $$G$$
 with $${m}_{t}\left(G\right)=3=es(G)$$
 are characterized. It is shown that for any three positive integers $$r, d$$
 and $$k\ge 6$$
 with $$r<d$$
 , there exists an extreme-support total monophonic graph $$G$$
 with monophonic radius $$r$$
 , monophonic diameter $$d$$
 and total monophonic number $$k$$
 . Also, for any three positive integers $$d, k$$
 and $$p$$
 with $$2\le d\le p-5$$
 and $$5\le k\le p-2$$
 and $$p-d-k\ge 0$$
 , there exists an extreme-support total monophonic graph $$G$$
 of order $$p$$
 with monophonic diameter $$d$$
 and $${m}_{t}\left(G\right)=k$$
 .

Extreme-Support Total Monophonic Graphs

For a connected graph [Formula: see text] of order at least two, a connected outer connected geodetic set [Formula: see text] of [Formula: see text] is called a minimal connected outer connected geodetic set if no proper subset of [Formula: see text] is a connected outer connected geodetic set of [Formula: see text]. The upper connected outer connected geodetic number [Formula: see text] of [Formula: see text] is the maximum cardinality of a minimal connected outer connected geodetic set of [Formula: see text]. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for any two integers [Formula: see text], [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the connected outer connected geodetic number of a graph. Also, another parameter forcing connected outer connected geodetic number [Formula: see text] of a graph [Formula: see text] is introduced and several interesting results on this parameter are studied.

K. Ganesamoorthy

Papers

On the monophonic number of a graph

The Connected Monophonic Number of a Graph

The total detour monophonic number of a graph

Extreme-Support Total Monophonic Graphs

The Upper and Forcing Connected Outer Connected Geodetic Numbers of a Graph