Generalized double bananas

Definition 42 (Generalized double banana)

For \(d\geq 3\) and \(2\leq t\leq d-1\), the graph \(B_{d,t}\) is defined by putting \(B_{d,t}=(G_1\cup G_2)-e\) where \(G_i\cong K_{d+2}\), \(G_1\cap G_2\cong K_{t}\) and \(e\in E(G_1\cap G_2)\). Note that the graph \(B_{3,2}\) is the well known flexible \(\mathcal{R}_3\)-circuit, commonly referred to as the double banana.

PyRigi: DoubleBanana()

The family \(\mathcal{B}_{d,d-1}^+\) consists of all graphs of the form \((G_1\cup G_2)-\{e,f,g\}\) where: \(G_1\cong K_{d+3}\) and \(e,f,g\in E(G_1)\); \(G_2\cong K_{d+2}\) and \(e\in E(G_2)\); \(G_1\cap G_2\cong K_{d-1}\); \(e,f,g\) do not all have a common end-vertex; if \(\{f,g\}\subset E(G_1)\setminus E(G_2)\) then \(f,g\) do not have a common end-vertex.

Theorem 24

Suppose \(G\) is a flexible \(\mathcal{R}_d\)-circuit with at most \(d+6\) vertices. Then either

• \(d=3\) and \(G\in \{B_{3,2}\}\cup \mathcal{B}_{3,2}^+\) or

• \(d\geq 4\) and \(G\in \{B_{d,d-1}\), \(B_{d,d-2}\}\cup \mathcal{B}_{d,d-1}^+\).

References: [GGJN22]

Definition 43 (t-sum)

Given three graphs \(G=(V,E)\), \(G_1=(V_1,E_1)\), and \(G_2=(V_2,E_2)\), we say that \(G\) is a \(t\)-sum of \(G_1,G_2\) along an edge \(e\) if \(G=(G_1\cup G_2)-e\), \(G_1\cap G_2=K_t\) and \(e\in E_1\cap E_2\).

Lemma 2

Suppose that \(G=(V,E)\) is the \(2\)-sum of \(G_1=(V_1,E_1)\) and \(G_2=(V_2,E_2)\). Then \(G\) is an \(\mathcal{R}_d\)-circuit if and only if \(G_1\) and \(G_2\) are both \(\mathcal{R}_{d}\)-circuits.

References: [GGJN22]

Using this lemma we can create one family of generalized bananas in which every element of the family is a \(\mathcal{R}_d\)-circuit.

A different generalization is as follows.

Let \(\mathcal{M}=(E,r)\) be a matroid with finite ground set \(E\) and rank function \(r\). A circuit of \(\mathcal{M}\) is a set \(C\subseteq E\) such that \(r(C)=|C|-1=r(C-e)\) for all \(e\in E\). Jackson, Nixon and Smith [JNS24] introduced a generalization to \(k\)-fold circuits i.e. sets \(D\subseteq E\) such that \(r(D)=|D|-2=r(D-e)\) for all \(e\in D\), for some fixed integer \(k\geq 0\).

Lemma 3

Let \(k\geq 1\) be an integer and let \(G\) be the graphical 2-sum of two graphs \(G_1\) and \(G_2\) along an edge \(e\). Suppose that \(e\) is not a coloop in either \(\mathcal{R}_d(G_1)\) or \(\mathcal{R}_d(G_2)\). Then \(G\) is a \(k\)-fold circuit in \(\mathcal{R}_d\) if and only if \(G_1\) is a \(k_1\)-fold \(\mathcal{R}_d\)-circuit and \(G_2\) is a \(k_2\)-fold \(\mathcal{R}_d\)-circuit for some \(k_1,k_2\geq 1\) with \(k_1+k_2=k+1\).

References: [JNS24]

Definition 44

Define \(\overline B_{d,d-1}\) to be obtained from \(B_{d,d-1}\) by adding back the edge \(e\). It follows that \(\textrm{cl} (B_{d,d-1})=\overline B_{d,d-1}\), and so \(\overline B_{d,d-1}\) is a flexible 2-fold circuit in \(\mathcal{R}_d\). We define the triple banana \(B^{(3)}_{d,d-1}\) to be the \(2\)-sum of \(\overline B_{d,d-1}\) and \(K_{d+2}\) again along \(e\). By Lemma, \(B^{(3)}_{d,d-1}\) is a 2-fold circuit in \(\mathcal{R}_d\). Iterating this process, we get that the \((k+1)\)-tuple banana \(B^{(k+1)}_{d,d-1}\) is a \(k\)-fold circuit in \(\mathcal{R}_d\).