On parametric equivalence, isomorphism and uniqueness: Cycle related graphs

1. Introduction

Unless stated otherwise, graphs will be finite, undirected and connected simple graphs. A shortest path having end vertices \(u\) and \(v\) is denoted by, \(u-v_{(in~G)}\). If \(d_G(u,v)\geq 2\) then a vertex \(w\) on \(u-v_{(in~G)}\), \(w\neq u\), \(w\neq v\) is called an internal vertex on \(u-v_{(in~G)}\). When the context is clear the notation such as \(d_G(u,v)\), \(deg_G(v)\) can be abbreviated to \(d(u,v)\), \(deg(v)\) and so on. Good references to important concepts, notation and graph parameters can be found in [1,2,3].

The notions of parametric equivalence, isomorphism and uniqueness had been introduced in [4]. For ease of reference we recall from [4] as follows: Let \(\rho\) denote some minimum or maximum graph parameter related to subsets \(V(G)\) of graph \(G\). Vertex subsets \(X\) and \(Y\) is said to be parametric equivalent or \(\rho\)-equivalent if and only if both \(X\), \(Y\) satisfy the parametric conditions of \(\rho\). This relation is denoted by \(X\equiv_\rho Y\). Furthermore, if \(X\equiv_\rho Y\) and the induced graphs \(\langle V(G)\backslash X\rangle \cong \langle V(G)\backslash Y\rangle\) then \(X\) and \(Y\) are said to be parametric isomorphic. This isomorphic relation is denoted by \(X \cong_\rho Y\). Let all possible vertex subsets of graph \(G\) which satisfy \(\rho\) be \(X_1, X_2, X_3,\dots, X_k\). If \(X_1 \cong_\rho X_2 \cong_\rho X_3 \cong_\rho \cdots \cong_\rho X_k\) then \(X_i\), \(1\leq i \leq k\) are said to be parametric unique or \(\rho\)-unique. The graph \(G\) is said to have a parametric unique or \(\rho\)-unique solution (or parametric unique \(\rho\)-set). If \(G\) has a unique (exactly one) \(\rho\)-set \(X\), then \(X\) is a parametric unique \(\rho\)-set.

This paper furthers the introductory research presented in [4].

2. Confluence in graphs

Shiny et al., [5] introduced the concept of a confluence set (a subset of vertices) of a graph \(G\), also see [6] for results on certain derivative graphs. Recall that for a non-complete graph \(G\), a non-empty subset \(\mathcal{X}\subseteq V(G)\) is said to be a confluence set if for every unordered pair \(\{u,v\}\) of distinct vertices (if such exist) in \(V(G)\backslash \mathcal{X}\) for which \(d_G(u,v)\geq 2\) there exists at least one \(u-v_{(in~G)}\) with at least one internal vertex, \(w\in \mathcal{X}\). Also a vertex \(u\in \mathcal{X}\) is called a confluence vertex of \(G\). A minimal confluence set \(\mathcal{X}\) (also called a \(\zeta\)-set) has no proper subset which is a confluence set of \(G\). The cardinality of a minimum confluence set is called the confluence number of \(G\) and is denoted by \(\zeta(G)\). A minimal confluence set is denoted by \(\mathcal{C}\). To distinguish between different graphs the notation \(\mathcal{C}_G\) may be used for a minimum confluence set of \(G\). We recall two important results from [4]. We remind that for a complete graph the confluence number is \(0\) hence, \(\mathcal{C}_{K_n}=\emptyset\), \(n\geq 1\).

Proposition 1. [4] A path \(P_n\) has a parametric unique \(\zeta\)-set if and only if \(n=1,2\) or \(n=4+3i\) or \(n=5+3i\), \(i=0,1,2,\dots\).

Proposition 2. [4] A cycle \(C_n\) has a parametric unique \(\zeta\)-set if and only if \(n=3,4\) or \(n=5+3i\) or \(n=6+3i\), \(i=0,1,2,\dots\).

2.1. Cycle related graphs

Henceforth, a cycle \(C_n\), \(n\geq 3\) of order \(n\) has the vertex set \(V(C_n)=\{v_i:i=1,2,3,\dots,n\}\).

• (a)   A wheel graph (simply, a wheel) \(W_n\) is obtained from a cycle \(C_n\), \(n\geq 3\) with an additional central vertex \(v_0\) and the additional edges \(v_0v_1\), \(1\leq i\leq n\). The cycle is called the rim and the edges \(v_0v_i\), \(1\leq i\leq n\) are called spokes. Alternatively, \(W_n=C_n+K_1\) and \(V(K_1)=\{v_0\}\).

Proposition 3. A wheel graph \(W_n\) has a parametric unique \(\zeta\)-set.

Proof. Since \(W_3\) is complete the result is trivial. For \(n\geq 4\) the distance \(d(v_i,v_j)\leq 2\) for all distinct pairs. For \(i,j\neq 0\) and \(v_i\) not adjacent to \(v_j\) there exists a \(3\)-path (or \(2\)-distance path) with \(v_0\) the internal vertex. Hence, the unique \(\zeta\)-set is \(\{v_0\}\), therefore parametric unique.

• (b)   A helm graph \(H_n\) is obtained from a wheel graph \(W_n\) by adding a pendent vertex (or leaf) \(u_i\) to each rim vertex \(v_i\).

Proposition 4.

• (a)   The helm graph \(H_3\) does not have a parametric unique \(\zeta\)-set.
• (b)   A helm graph \(H_n\), \(n\geq 4\) has a parametric unique \(\zeta\)-set.

Proof.

• (a)   Consider \(H_3\). Clearly and without loss of generality the sets \(X_1=\{v_0,v_1,v_2\}\), \(X_2 = \{v_1,v_2,v_3\}\) and \(X_3=\{v_1,v_2,u_3\}\) are all minimal confluence sets. Hence \(\zeta(H_3)\leq 3\). It is easy to verify that no \(2\)-vertex subset is a confluence set. Thus, \(\zeta(H_3)>2\). Also, \(\langle V(H_3)\backslash X_1\rangle \ncong \langle V(H_3)\backslash X_2\rangle\). Therefore \(H_3\) does not have a parametric unique \(\zeta\)-set. The aforesaid follows in essence from the fact that \(H_3\) is complete. Therefore, it is not necessary for \(v_0\) to be in all \(\zeta\)-sets.
• (b)   For \(H_n\), \(n\geq 4\) the distance \(d(u_i,u_{i+1})=3\) hence a rim vertex is required. The distance \(d(u_i,u_{i+2})=5\) hence the vertex \(v_0\) will suffice along the \(5\)-path \(u_iv_iv_0v_{i+2}u_{i+2}\). By symmetry considerations and therefore up to isomorphism and without loss of generality we have two subcases.

  Subcase 1. If \(n\) is even the set \(X_1=\{v_0, v_1,v_3,v_5,\dots,v_{n-1}\}\) is a \(\zeta\)-set and clearly \(H_n\) has a parametric unique \(\zeta\)-set.

  Subcase 2. If \(n\) is odd the sets \(X_1=\{v_0, v_1,v_3,v_5,\dots,v_{n-1}\}\) and \(X_2=\{v_0, v_1,v_3,v_5,\dots,v_{n-2},v_n\}\) are a \(\zeta\)-sets. Clearly \(\langle V(H_n)\backslash X_1\rangle \cong \langle V(H_n)\backslash X_1\rangle\). Thus \(H_n\) has a parametric unique \(\zeta\)-set.

As a direct consequence of the proof of Proposition 4, we get the next corollary.

Corollary 1. A helm graph has \(\zeta(H_n)=\lceil \frac{n}{2}\rceil +1\).

• (c)   A flower graph \(Fl_n\) is obtained from a helm graph \(H_n\) by adding the edges \(v_0u_i\), \(1\leq i\leq n\).

Proposition 5. A flower graph \(Fl_n\) has a parametric unique \(\zeta\)-set.

Proof. The result follows by similar reasoning as in the proof of Proposition 3.

As a direct consequence of Proposition 5, we get the next corollary.

Corollary 2. A flower graph has \(\zeta(Fl_n)=1\).

• (d)   A closed helm graph \(H^c_n\) is obtained from a helm graph \(H_n\) by completing a cycle, \(C'_n=u_1u_2u_3\cdots u_nu_1\) on the leafs of \(H_n\).

Proposition 6.

• (a)   A closed helm graph \(H^c_n\) for \(n=4\) or \(n\) is odd does not have a parametric unique \(\zeta\)-set.
• (b)   A closed helm graph \(H^c_n\), \(n \geq 6\) and even, has a parametric unique \(\zeta\)-set.

Proof. It is easy to verify that all distance paths such that \(d(u_i,u_j)\leq 3\) are paths on \(C'_n\). Also, for \(u_i,u_j\in \mathcal{C}_{C'_n}\) we have \(d(u_i,u_j)\leq 3\). It follows that \(\mathcal{C}_{C'_n}\subseteq \mathcal{C}_{H^c_n}\).

• (a)   By similar reasoning to that in the proof of Proposition 4(a) it follows that \(H^c_3\) and \(H^c_4\) do not have a unique \(\zeta\)-set.

  From the set \(X_1=\{v_i: u_i\notin \mathcal{C}_{C'_n}\}\cup \{v_0\}\) it is possible to select a minimum confluence set in respect of the spanning subgraph \(H_n\) say set \(X_2\). The set \(\mathcal{C}_{H^c_n}=\mathcal{C}_{C'_n}\cup X_2\) is a minimum confluence set.

  Subcase (a)(1). Since by symmetry the choice of say, \(X_2\) can be fixed, For \(n\geq 5\) and odd, the choice of \(\mathcal{C}_{C'_n}\) can rotate such that \(\langle V(H^c_n)\backslash \mathcal{C}_{H^c_n}\rangle\) does not remain isomorphic.

• (b)   By similar reasoning \(X_2\) can be fixed. However, for \(n\geq 6\) and even and by symmetry properties of \(C'_n\) all choices of \(\mathcal{C}_{C'_n}\) yield isomorphic \(\langle V(H^c_n)\backslash \mathcal{C}_{H^c_n}\rangle\).

As a direct consequence of the proof of Proposition 6, we get the next corollary.

Corollary 3. A closed helm graph has \(\zeta(H^c_n)= \lceil \frac{n}{2}\rceil +1\).

• (e)   A gear graph \(G_n\) is obtain from a wheel graph \(W_n\) by inserting a vertex \(u_i\) on the edge \(v_iv_{i+1}\) and \(n+1\equiv 1\). Note that \(G_n\) has \(2n+1\) vertices and \(3n\) edges. The rim is now called a boundary cycle denoted by \(C^b(G_n)\).

Proposition 7.

• (a)   \(G_3\) has a parametric unique \(\zeta\)-set.
• (b)   A gear graph \(G_n\) and \(n \geq 5\) is odd does not have a parametric unique \(\zeta\)-set.
• (c)   A gear graph \(G_n\) and \(n \geq 4\) is even has a parametric unique \(\zeta\)-set.

Proof.

• (a)   For \(G_3\) it follows easily that up to isomorphism the \(\zeta\)-set \(\{u_1,v_3\}\) is unique.
• (b)   The inner-area enclosed by the cycle \(C'_{2n} =v_1u_1v_2u_2\cdots v_nu_nv_1\) can be partitioned into \(n\) planar areas, each enclosed by a \(C_4\). For all pairs \(v_i,v_j\) it is necessary and sufficient that \(v_0\in \zeta\)-set. Let \(n \geq 5\) be odd. Without loss of generality, an optimal minimal confluence set is given by \(X_1=\{v_0,u_1,u_3,\dots, u_{n-2},u_{n-1}\}\) or \(X_2=\{v_0,u_1,u_3,\dots, u_{n-2},v_n\}\) or \(X_3=\{v_0,u_1,u_3,\dots, u_{n-2},u_n\}\). Hence, \(\zeta(G_n)\leq \lceil \frac{2n}{4}\rceil +1= \lceil \frac{n}{2}\rceil +1\). Because the boundary cycle \(C^b(G_n)\) has \(\zeta(C^b(G_n))= \lceil \frac{2n}{3}\rceil\) it follows that \(\zeta(G_n) \geq \lceil \frac{2n}{3}\rceil\). However for \(n\) is odd, \(\lceil \frac{2n}{3}\rceil = \lceil \frac{n}{2}\rceil +1\). Since, \[\langle V(G_n)\backslash X_1\rangle \ncong \langle V(G_n)\backslash X_2\rangle.\] It follows that a gear graph \(G_n\) does not have a parametric unique \(\zeta\)-set for \(n\) is odd.
• (c)   For \(n \geq 4\) and even, reasoning similar to that in (b) show that up to isomorphism the \(\zeta\)-set \(X_1=\{v_0,u_1,u_3,\dots, u_{n-2},u_{n-1}\}\) is unique. Reasoning in respect of bounds on \(\zeta(G_n)\) similar to that in (a) settles the result.

As a direct consequence of the proof of Proposition 7, we get the next corollary.

Corollary 4. The gear graph \(G_3\) has \(\zeta(G_3)=2\). A gear graph of order \(n\geq 4\) has \(\zeta(G_n)=\lceil \frac{n}{2}\rceil +1\).

• (f)   A sun graph \(S^\boxtimes_n\), \(n\geq 3\) is obtained by taking the complete graph \(K_n\) on the vertices \(v_1,v_2,v_3,\dots,v_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the edges \(v_iu_i\), \(u_iv_{i+1}\) and \(n+1\equiv 1\). The boundary cycle of a sun graph is the cycle \(C^b(S^\boxtimes_n) =v_1u_1v_2u_2v_3u_3\cdots u_nv_1\).

Proposition 8. A sun graph \(S^\boxtimes_n\), \(n\geq 3\) has a parametric unique \(\zeta\)-set if and only if \(C^b(S^\boxtimes_n)\) is of order \(n= 3i\), \(i=1,2,3,\dots\)

Proof. Since all pairs \(v_i,v_j\) are adjacent it suffices to only consider a \(\zeta\)-set of \(C^b(S^\boxtimes_n)\). Since \(deg(u_i)=2\) and \(deg(v_j)=3\) any \(\zeta\)-set must be graphically symmetrical for a sun graph to have a parametric unique \(\zeta\)-set. A graphically symmetrical \(\zeta\)-set means that, measured along the boundary cycle, \(min\{d(v_j,u_k):v_j,u_k\in \zeta\)-set\(\}=3\). It implies that \(n=3i\), \(i=1,2,3,\dots\).

The converse follows from the fact that sun graphs with \(C^b(S^\boxtimes_n)\) of order \(n\neq 3i\), \(i=1,2,3,\dots\) do not have graphically symmetrical \(\zeta\)-sets of even order.

Note that if a sun graph has a parametric unique \(\zeta\)-set then \(\zeta(S^\boxtimes_n)\) is even. Furthermore, as a direct consequence of the proof of Proposition 8, we get the next corollary.

Corollary 5. A sun graph has \(\zeta(S^\boxtimes_n)=\lceil \frac{2n}{3}\rceil\).

• (g)   A sunflower graph \(S^\circledast_n\), \(n\geq 3\) is obtained by taking the wheel graph \(W_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the edges \(v_iu_i\), \(u_iv_{i+1}\) and \(n+1\equiv 1\). The boundary cycle of a sun graph is the cycle \(C^b(S^\circledast_n) =v_1u_1v_2u_2v_3u_3\cdots u_nv_1\).

Proposition 9. A sunflower graph \(S^\circledast_n\), \(n\geq 3\) does not have a parametric unique \(\zeta\)-set.

Proof. For all pairs \(v_i,v_j\) it is sufficient that \(v_0\in \zeta\)-set. Thereafter any \(\zeta\)-set \(X_1\) in respect of \(C^b(S^\circledast_n)\) is required to obtain \(\mathcal{C}_{S^\circledast_n}= X_1\cup \{v_0\}\). It implies that \(\zeta(S^\circledast_n)= n\). In turn, the aforesaid confluence number permits that say, \(X_2=\{v_1,v_2,v_3,\dots,v_n\}\) or \(X_3=\{v_1,v_2,v_3,\dots,v_{n-1},u_{n-1}\}\) are \(\zeta\)-sets. Since, \(\langle V(S^\circledast_n)\backslash X_1\rangle \ncong \langle V(S^\circledast_n)\backslash X_2\rangle \ncong \langle V(S^\circledast_n)\backslash X_3\rangle\) the result follows.

As a direct consequence of the proof of Proposition 9, we get the next corollary.

Corollary 6. A sunflower graph has \(\zeta(S^\circledast_n)= n\).

• (h)   A sunlet graph \(S^\circleddash_n\), \(n\geq 3\) is obtained by taking cycle \(C_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the pendent edges \(v_iu_i\).

Proposition 10. A sunlet graph \(S^\circleddash_n\), \(n\geq 3\) has a parametric unique \(\zeta\)-set.

Proof. Case 1. Let \(n\geq 3\) and odd. Without loss of generality and by isomorphism, it is easy to verify that the sets \(X_1=\{v_1,v_3,v_5,\dots,v_n\}\) and \(X_2=\{v_1,v_3,v_5,\dots,v_{n-2},v_{n-1}\}\) are \(\zeta\)-sets. Furthermore, up to isomorphism those are the only distinguishable \(\zeta\)-sets. Since, \[\langle (V(S^\circleddash_n)\backslash X_1\rangle \cong \langle (V(S^\circleddash_n)\backslash X_2\rangle,\] the result follows for \(n\geq 3\) and odd.

Case 2. By similar reasoning as in Case 1 the result follows for \(n\geq 4\) and even.

As a direct consequence of the proof of Proposition 10, we get the next corollary.

Corollary 7. A sunlet graph has \(\zeta(S^\circleddash_n)=\lceil \frac{n}{2}\rceil\).

• (i)   A circular ladder (or prism graph) \(L^\circ_n\), \(n\geq 3\) is obtained by taking two cycles of equal order \(n\). Label as, \(C^1_n=v_1v_2v_3\cdots v_nv_1\) and \(C^2_n= u_1u_2u_3\cdots u_nu_1\). Add the edges \(v_iu_i\), \(1\leq i \leq n\). A circular ladder can be viewed as \(H^c_n-v_0\).

Proposition 11. A circular ladder graph \(L^\circ_n\) has a parametric unique \(\zeta\)-set if and only if \(n=4 \) or \(n=3i\) for \(i=2,3,4,...\).

Proof. Part 1. For \(n=4\), \(X_{i}=\{u_{i},v_{j}\}\), \(i=1,2,3,4\), \(j\in \{1,2,3,4\}\) such that \(d(u_{i},v_{j})=3\), are the minimum confluence sets for \(L^\circ_4\). Since \(\langle V(L^\circ_4)\backslash X_{i}\rangle\) are \(C_{6}\) for \(i=1,2,3,4\), we have the result for \(n=4\).

In a circular ladder graph \(L^\circ_n\), \(n\neq 4\) there are \(n\) copies of \(C_{4}=v_{i}u_{i}u_{i+1}v_{i+1}\). For each \(C_{4}=v_{i}u_{i}u_{i+1}v_{i+1}\), at least one of the vertices \(v_{i},u_{i},u_{i+1},v_{i+1}\) belongs to every minimum confluence set of \(L^\circ_n\).

Part 2. For \(n=3\), \(X_{1}=\{v_{1},v_{2}\}\) and \(X_{2}=\{v_{1},u_{2}\}\) are two minimum confluence set for \(L^\circ_3\). However, \(\langle V(L^\circ_3)\backslash X_{1}\rangle\) and \(\langle V(L^\circ_3) \backslash X_{2}\rangle\) are not isomorphic. Hence \(L^\circ_3\) has no unique parametric set.

Part 3. For \(n=3i\), \(i=2,3,..\), let \(\mathcal{C}_{C_{n}}(v_{i})\) be a minimum confluence set of \(C_{n}\) starting from \(v_{i}\) and \(C_{C_{n}^{'}}(u_{j})\) be a minimum confluence set of \(C_{n}^{'}\) starting from \(u_{j}\). Then for \(i \neq j\),\(X_{ij}= \mathcal{C}_{C_{n}}(v_{i}) \cup \mathcal{C}_{C_{n}^{'}}(u_{j})\) is a minimum confluence set for \(L^\circ_n\) and \(\langle V(L^\circ_n)\backslash X_{ij}\rangle\) consists of \(\frac{n}{3}\) copies of \(P_{3}\). Hence the result for \(n=3i\), \(i=2,3,..\).

Part 4. If \(n \equiv 2(mod~3)\). Let \(X_{1}\) be the minimum confluence set for \(L^\circ_n\) such that \(u_{i},u_{i+2},v_{i+1} \in X_{1}\) and let \(X_{2}\) be the minimum confluence set for \(L^\circ_n\) such that \(u_{i},u_{i+2},v_{i} \in X_{2}\). Then \(\langle V(L^\circ_n)\backslash X_{1}\rangle\) and \(\langle V(L^\circ_n)\backslash X_{2}\rangle\) are not isomorphic. Hence \(L^\circ_n\) has no parametric unique set if \(n_{\geq 5} \equiv 2(mod~3)\).

By a similar argument we have to prove that \(L^\circ_n\) has no parametric unique set if \(n_{\geq 7} \equiv 1(mod~3)\).

Since all \(n\in \mathbb{N}_{\geq 3}\) have been accounted for the 'if' has been settled.

For all valid cases the converse, 'only if', follows through reasoning by contradiction.

Corollary 8. A circular ladder has, \begin{equation*} \zeta(L^\circ_n) = \begin{cases} 2, & \mbox{if \(n= 4\)};\\ 2\lceil \frac{n}{3}\rceil, & \mbox{if \(n=3\) or \(n \geq 5\)}.\\ \end{cases} \end{equation*}

Proof. The result is a consequence of the proof of Proposition 11. The exception lies in the fact that \(L^\circ_4\) has \(5=n_{=4}+1\) cycles \(C_4\) to account for. All other \(L^\circ_{n_{\neq 4}}\) have \(n\) cycles \(C_4\) to account for.

Observe that the confluence number of a circular ladder is always even.

• (j)   A tadpole graph \(T(m,n)\), \(m\geq 3\), \(n\geq 1\) is obtained from a cycle \(C_m=v_1v_2v_3\cdots v_mv_1\) and a path \(P_n = u_1u_2u_3\cdots u_n\) by adding an edge between an end-vertex of \(P_n\) and a vertex of \(C_m\). The new edge is also called a bridge.

Proposition 12. A tadpole graph \(T(m,n)\), \(m\geq 3\), \(n\geq 1\):

• (a)   Tadpole graphs \(T(3,n)\), \(n\geq 1\) have a parametric unique \(\zeta\)-set if and only if \(n=3i\), \(i=1,2,3,\dots\).
• (b)   Tadpole graphs \(T(4,1)\), \(T(4,2)\) have a parametric unique \(\zeta\)-sets.
• (c)   Tadpole graphs \(T(5,1)\) does not have a parametric unique \(\zeta\)-set and \(T(5,2)\) has.
• (d)   Tadpole graphs \(T(m,1)\), \(T(m,2)\), \(m\geq 6\) have a parametric unique \(\zeta\)-set if and only if \(m=6+3i\), \(i= 0,1,2,\dots\)
• (e)   Tadpole graphs \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) have a parametric unique \(\zeta\)-set if and only if both the cycle \(C_m\) and the path \(P_n\) have parametric unique \(\zeta\)-sets.
• (f)   All other tadpole graphs as excluded through (a) to (f) do not have a parametric unique \(\zeta\)-set.

Proof.

• (a)   The tadpole graphs \(T(3,n)\), \(n\geq 1\) does not have a parametric unique \(\zeta\)-set for \(P_1\), \(P_2\) (straightforward).

  Subcase (a)(1). For \(n+2=5+3i\), \(i=0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is unique hence, \(T(3,n)\) has a parametric unique \(\zeta\)-set.

  Subcase (a)(2). For \(n+2=6+3i\), \(i= 0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is not parametric unique hence, \(T(3,n)\) does not have a parametric unique \(\zeta\)-set.

  Subcase (a)(3). For \(n+2=7+3i\), \(i= 0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is parametric unique. However, since some \(\zeta\)-sets may contain vertex \(v_j\) of the bridge the tadpole \(T(3,n)\) does not have a parametric unique \(\zeta\)-set.

  All tadpoles \(T(3,n)\), \(n\geq 1\) have been accounted for because, \[ \mathbb{N}=\{1,2\}\cup \{3+3i:i=0,1,2,\dots\}\cup \{4+3i:i=0,1,2,\dots\}\cup \{5+3i:i=0,1,2,\dots\}. \]
• (b)   The tadpole graphs \(T(4,n)\), \(n\geq 1\) have a parametric unique \(\zeta\)-set for \(P_1\), \(P_2\). It follows from the fact that a bridge vertex say, \(v_i\) has to be in any \(\zeta\)-set.

  Subcases \(n+2=5+3i\), \(n+2=6+3i\) and \(n+2=7+3i\), \(i= 0,1,2,\dots\) will be settled in (d) and (e) below.

• (c)   The tadpole graphs \(T(5,n)\), \(n\geq 1\) does not have a parametric unique \(\zeta\)-set for \(P_1\) bacause it is easy to verify that an end-vertex of the bridge need not be in all \(\zeta\)-sets. However for \(P_2\) the tadpole has a parametric unique \(\zeta\)-set. It follows from the fact that a bridge vertex say, \(v_i\) has to be in any \(\zeta\)-set.

  Subcases \(n+2=5+3i\), \(n+2=6+3i\) and \(n+2=7+3i\), \(i= 0,1,2,\dots\) will be settled in (d) and (e) below.

• (d)   The tadpoles \(T(m,1)\), \(T(m,2)\), \(m\geq 6\) do not require that vertices \(u_1\) and/or \(u_2\) to necessarily be in a \(\zeta\)-set. Hence, all \(\zeta\)-sets of cycle \(C_m\) which contain a vertex of the bridge suffice to be \(\zeta\)-sets of the tadpoles. Therefore has a parametric unique \(\zeta\)-set if and only if \(C_m\) has a unique \(\zeta\)-set. Therefore, if and only if \(m=6+3i\), \(i= 0,1,2,\dots\) The converse follows easily by contradiction.
• (e)   Finally, for a tadpole \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) and both the cycle \(C_m\) and the path \(P_n\) have parametric unique \(\zeta\)-sets, it is easy to verify that the \(\zeta\)-sets of the tadpole all contain a vertex \(v_j\) of the bridge. Therefore the tadpole has a parametric \(\zeta\)-set. Else, it is always possible to find a \(\zeta\)-set of the tadpole which contains a vertex \(v_j\) which is on the bridge and another \(\zeta\)-set which does not. Therefore, such tadpoles do not have a parametric unique \(\zeta\)-set. Hence, the tadpoles \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) have a parametric unique \(\zeta\)-set if and only if both \(C_m\) and \(P_n\) have parametric unique \(\zeta\)-sets.
• (f)   All other tadpole graphs which were excluded through reasoning of proof, (a) to (e) do not have a parametric unique \(\zeta\)-set.

• (k)   A lollipop graph \(L^\boxtimes(m,n)\), \(m\geq 3\), \(n\geq 1\) is obtained from a complete graph \(K_m\) and a path \(P_n\) by adding a bridge between an end-vertex of \(P_n\) and a vertex of \(C_m\).

Proposition 13. A lollipop graph \(L^\boxtimes(m,n)\), \(m\geq 3\), \(n\geq 1\) has a parametric unique \(\zeta\)-set if and only if \(n=3i\), \(i=1,2,3,\dots\).

Proof. The proof follows directly from the proof of Proposition 12(a).

• (l)   A generalized barbell graph \(B(n,m)\), \(n,m\geq 3\) is obtained from two complete graph \(K_n\), \(K_m\) and adding a bridge.

Proposition 14. A generalized barbell graph \(B(n,m)\), \(n,m\geq 3\) has a parametric unique \(\zeta\)-set if and only if \(n=m\).

Proof. Let \(K_n\) be on vertices \(v_1,v_2,v_3,\dots,v_n\) and \(K_m\) on vertices \(u_1,u_2,u_3,\dots,u_m\). For any pair \(v_iu_j\) and edge \(v_iu_j\) not the bridge, the distance \(d(v_i,u_j)=2\) or \(3\). Therefore any vertex of the bridge yields a \(\zeta\)-set. Without loss of generality let the \(\zeta\)-set be \(\{v_k\}\). It follows that \(\langle V(B(n,m))\backslash \{v_k\}\rangle \cong K_{n-1}\cup K_m\). Hence, \(B(n,m)\) has a parametric unique \(\zeta\)-set if and only if \(n=m\).

3. Conclusion

The study of cycle related graphs has not exhausted. Note that for those cycle related graphs which do not have a parametric unique \(\zeta\)-set the proof by contradiction can be utilized well.

The idea of combined parametric conditions remains open. Note that the parametric conditions will be ordered pairs. For example, the path \(P_3 = v_1v_2v_3\) has a unique minimum dominating set i.e. the \(\gamma\)-set \(X_1=\{v_2\}\). Since \(X_1\) is also a \(\zeta\)-set of \(P_3\) the set is said to be a parametric unique \((\gamma,\zeta)\)-set. However, since \(X_1\) per se is not a parametric unique \(\zeta\)-set, it cannot be said to be a parametric unique \((\zeta,\gamma)\)-set. On the other hand for a star \(S_{1,n}\), \(n\geq 3\) the set \(X_1=\{v_0\}\) is both a parametric \((\gamma,\zeta)\)-set and a parametric unique \((\zeta,\gamma)\)-set. Studying such parametric combinations for say parameters \(\rho_1(G)\) and \(\rho_2(G)\) requires that, \(\rho_1(G)=\rho_2(G)\).

Conjecture 1. If graph \(G\) has a pendent vertex then \(G\) has a unique \(\zeta\)-set if and only no \(\zeta\)-set exists which contains a pendent vertex.

A strict proof of Corollary 8 through mathematical induction is an interesting exercise for the reader.

Acknowledgments

The authors would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.