Two-connected signed graphs with maximum nullity at most two

Let $v_{1},v_{2},v_{3},v_{4}$ be the vertices on the rim of $W_{4}$ in this cyclic order, and let $v$ be the hub of $W_{4}$. We assume that the edges between $v_{1}$ and $v_{2}$ and between $v_{3}$ and $v_{4}$ are even. Suppose for a contradiction that $M(W_{4}^{o})\geq 3$. Then there exists a matrix $A=[a_{i,j}]\in S(W_{4}^{o})$ with nullity $\geq 3$. Since $\mbox{nullity}(A)\geq 3$, there exist nonzero vectors $x,y\in\ker(A)$ with $x_{v_{1}}=x_{v_{2}}=0$ and $y_{v_{1}}=y_{v_{4}}=0$. If $x_{v}=0$, then from $Ax=0$ it would follow that $x_{v_{4}}=x_{v_{3}}=0$. This contradiction shows that $x_{v}\not=0$. We may assume that $x_{v}>0$. Then, since the edge between $v$ and $v_{1}$ is even, the edge between $v_{1}$ and $v_{4}$ is odd, and $Ax=0$, it follows that $x_{v_{4}}>0$. Similarly, $x_{v_{3}}>0$. Let $a_{v}$ denote the row of $A$ corresponding to $v$. Then $a_{v}x=0$, and, since $x_{v}>0$, $x_{v_{4}}>0$, $x_{v_{3}}>0$, it follows that $a_{v,v}>0$.

We will now do the same with the vector $y$. If $y_{v}=0$, then it would follow that $y_{v_{2}}=y_{v_{3}}=0$. This contradiction shows that $y_{v}\not=0$. We may assume that $y_{v}>0$. Then, since the edge between $v_{1}$ and $v_{2}$ is even, the edge between $v_{1}$ and $v$ is even, and $Ay=0$, it follows that $y_{v_{2}}<0$. Similarly, $y_{v_{3}}<0$. Since $a_{v}y=0$ and $y_{v}>0$, $y_{v_{2}}<0$, $y_{v_{3}}<0$, it follows that $a_{v,v}<0$. We have arrived at a contradiction, and we can conclude that $\xi(W_{4}^{o})\leq M(W_{4}^{o})\leq 2$. Since $W_{4}^{o}$ contains an odd cycle, $M(W_{4}^{o})\geq\xi(W_{4}^{o})\geq\xi(K_{2}^{=})=2$, and hence $M(W_{4}^{o})=\xi(W_{4}^{o})=2$. ∎

Since $G$ has a $K_{4}$-minor and all vertices of $K_{4}$ have degree three, $G$ has a subgraph isomorphic to a subdivision of $K_{4}$. Hence there are distinct vertices $v_{1},v_{2},v_{3},v_{4}$ and openly disjoint paths $P_{1},\ldots,P_{6}$ of length $\geq 1$ in $G$, where $P_{1}$ has ends $v_{1}$ and $v_{2}$, $P_{2}$ has ends $v_{1}$ and $v_{3}$, $P_{3}$ has ends $v_{1}$ and $v_{4}$, $P_{4}$ has ends $v_{2}$ and $v_{3}$, $P_{5}$ has ends $v_{2}$ and $v_{4}$, and $P_{6}$ has ends $v_{3}$ and $v_{4}$. Let $H$ be the subgraph spanned by $v_{1},v_{2},v_{3},v_{4}$ and $P_{1},\ldots,P_{6}$. Since $(G,\Sigma)$ has no weak $K_{4}^{e}$- or $K_{4}^{o}$-minor, $(H,\Sigma\cap E(H))$ has no weak $K_{4}^{e}$- or $K_{4}^{o}$-minor.

By re-signing if needed, we may assume that, in $(G,\Sigma)$, $P_{1}$ is an odd path and that $P_{2},\ldots,P_{6}$ are even paths. We can see that the paths $P_{1}$, $P_{2}$, $P_{5}$, and $P_{6}$ consist of single edges as follows. $P_{6}$ is a single edge, for otherwise, possibly after re-signing, we contract an odd edge in $P_{6}$ and obtain a signed graph that contains $K_{4}^{o}$ as a weak minor. $P_{1}$ is a single edge, for otherwise, we contract an odd edge in $P_{1}$ and obtain a signed graph that contains $K_{4}^{e}$ as a weak minor. If both $P_{3}$ and $P_{5}$ have length $\geq 2$, then, possibly after re-signing, we contract an odd edge in $P_{3}$ and in $P_{5}$, and obtain a signed graph that has $K_{4}^{o}$ as a weak minor. Hence at least one of $P_{3}$ and $P_{5}$ consists of a single edge. In the same way, at least one of $P_{2}$ and $P_{4}$ consists of a single edge. If both $P_{2}$ and $P_{3}$ have length $\geq 2$, then possibly after re-signing, we contract an odd edge in $P_{2}$ and in $P_{3}$, and obtain a signed graph that has $K_{4}^{o}$ as a weak minor. Hence at least one of $P_{2}$ and $P_{3}$ consists of a single edge. In the same way, at least one of $P_{4}$ and $P_{5}$ consists of a single edge. If one of $P_{2}$ and $P_{5}$ has length $\geq 2$, then $P_{3}$ and $P_{4}$ consist each of a single edge. Hence, we can conclude that both $P_{2}$ and $P_{5}$ consist of single edges, or that both $P_{3}$ and $P_{4}$ consist of single edges. By symmetry we may assume that $P_{2}$ and $P_{5}$ consist of single edges.

Since $G$ has no $W_{4}$-minor, each path connecting a vertex of $P_{3}$ to a vertex of $P_{4}$ must contain at least one vertex of $\{v_{1},v_{4}\}$ and at least one vertex of $\{v_{2},v_{3}\}$. Thus $(G,\Sigma)$ has a wide separation. ∎

Since $G$ has a $K_{2,3}$-minor, $G$ has a subgraph $H$ isomorphic to a subdivision of $K_{2,3}$. Hence there are vertices $v_{1},v_{2},v_{3},v_{4},v_{5}$ and openly disjoint paths $P_{1},\ldots,P_{6}$ of length $\geq 1$ in $H$, where $P_{1}$ has ends $v_{1}$ and $v_{2}$, $P_{2}$ has ends $v_{1}$ and $v_{3}$, $P_{3}$ has ends $v_{1}$ and $v_{4}$, $P_{4}$ has ends $v_{2}$ and $v_{5}$, $P_{5}$ has ends $v_{3}$ and $v_{5}$, and $P_{6}$ has ends $v_{4}$ and $v_{5}$. We now view the paths $P_{1},\ldots,P_{6}$ as paths in the signed graph $(H,\Sigma\cap E(H))$. As $(G,\Sigma)$ has no weak $K_{2,3}^{e}$-minor, $(G,\Sigma)$ has a weak $K_{2,3}^{i}$-minor. Hence we may re-sign $(H,\Sigma\cap E(H))$ such that $P_{1}$ is odd and $P_{2},\ldots,P_{6}$ are even.

$P_{1}$ is a single edge, for otherwise, possibly after re-signing, we contract an odd edge in $P_{1}$ and obtain a signed graph that contains $K_{2,3}^{e}$ as a weak minor. Similarly, $P_{4}$ is a single edge. If both $P_{2}$ and $P_{3}$ have length $\geq 2$, then, possibly after re-signing, we contract an odd edge in $P_{2}$ and in $P_{3}$, and obtain a signed graph that has $K_{2,3}^{e}$ as a weak minor. Hence, at least one of $P_{2}$ and $P_{3}$ consists of a single edge. Similarly, at least one of $P_{5}$ and $P_{6}$ consists of a single edge, at least one of $P_{2}$ and $P_{6}$ consists of a single edge, and at least one of $P_{3}$ and $P_{5}$ consists of a single edge. Hence at most one path of $P_{2},P_{3},P_{5},P_{6}$ has length $\geq 2$. By symmetry, we may assume that each path of $P_{2},P_{3},P_{5}$ consists of a single edge. Let $Q$ be the concatenation of $P_{3}$ and $P_{6}$. Since $G$ has no $K_{4}$-minor, there are no paths in $G$ connecting $v_{2}$ and $v_{3}$, there are no paths in $G$ connecting $v_{2}$ and an internal vertex of $Q$, and there are no paths in $G$ connecting $v_{3}$ and an internal vertex of $Q$. Hence, $(G,\Sigma)$ has a wide separation $[G_{1},G_{2}]$, where $G_{1}$ is isomorphic to $K_{2}^{c}$. ∎

If at most one triangle of $(W_{4},\Sigma)$ is even, then $(W_{4},\Sigma)$ has a weak $K_{4}^{o}$-minor. If at most one triangle of $(W_{4},\Sigma)$ is odd, then $(W_{,}\Sigma)$ has a weak $K_{4}^{e}$-minor. So we may assume that $(W_{4},\Sigma)$ has exactly two odd triangles. If they share an edge, then $(W_{4},\Sigma)$ has a weak $K_{2,3}^{e}$-minor. If they do not share an edge, then $(W_{4},\Sigma)$ is sign-equivalent to $W_{4}^{o}$. ∎

Let $v_{1},v_{2},v_{3},v_{4}$ be the vertices on the rim of $W_{4}$ in this order. Up to symmetry there is only one possibility to add an edge between two nonadjacent vertices in $W_{4}$. We may assume that we add the edge $e$ between $v_{1}$ and $v_{3}$. If $e$ is an even edge, then the resulting signed graph has a weak $K_{4}^{o}$-minor. If $e$ is an odd edge, then the resulting graph has a weak $K_{4}^{e}$-minor. ∎

If $G$ has a vertex of degree two, then $G$ arises from $W_{4}$ by inserting a new vertex on an edge $e$ of $W_{4}$, which results in a path $P$ of length two in $G$. If $e$ is an even edge on the rim of $W_{4}$, then, possibly after re-signing, we contract all but one odd edge of $P$. The resulting signed graph contains a weak $K_{4}^{o}$-minor. If $e$ is an odd edge on the rim of $W_{4}$, then, possibly after re-signing, we contract all but one even edge of $P$. The resulting signed graph contains a weak $K_{4}^{e}$-minor. If $e$ is a spoke of $W_{4}$, then, possibly after re-signing, we contract all but one odd edge of $P$. The resulting signed graph contains a weak $K_{4}^{o}$-minor.

So we may assume that $G$ has no vertex of degree two. Then $G$ is isomorphic either to the prism, $C_{6}^{c}$ (see Figure 3), or to $K_{3,3}$. Let $e$ be the edge in $G$ such that contracting $e$ yields $W_{4}$.

If $G$ is isomorphic to $K_{3,3}$, then $(G,\Sigma)-e$ contains a weak $K_{4}^{o}$-minor. Suppose next that $G$ is isomorphic to the prism. Then $(G,\Sigma)$ has either two odd triangles or two even triangles. If $(G,\Sigma)$ has two odd triangles, then, possibly after re-signing, we contract $e$ and another edge not on the two odd triangles, and obtain a signed graph that contains a $K_{4}^{o}$ as a signed subgraph. If $(G,\Sigma)$ has two even triangles, then, possibly after re-signing, we contract $e$ and another edge not on the two even triangles, and obtain a signed graph that contains a $K_{4}^{e}$ as a signed subgraph. ∎

Suppose for a contradiction that there exists a signed graph $(G,\Sigma)$ such that $G$ has a $W_{4}$-minor, but $(G,\Sigma)$ has no weak $K_{4}^{o}$-, $K_{4}^{e}$-, or $K_{2,3}^{e}$-minor and it is not the case that the edges in each parallel class of $(G,\Sigma)$ have the same parity and, after removing from each parallel class all but one edge, we obtain $W_{4}^{o}$. We take $(G,\Sigma)$ with $|V(G)|+|E(G)|$ as small as possible.

If the underlying simple graph of $G$ is isomorphic to $W_{4}$, then there must be parallel edges of different parity in $(G,\Sigma)$. In this case $(G,\Sigma)$ has a weak $K_{4}^{o}$-, $K_{4}^{e}$-, or $K_{2,3}^{e}$-minor by Lemma 8. So we may assume that the underlying simple graph of $G$ is not isomorphic to $W_{4}$. Since $G$ has a $W_{4}$-minor and $G$ is connected, there exists a signed graph $(H,\Omega)$ such that $(H,\Omega)$ is a minor of $(G,\Sigma)$, the underlying simple graph of $H$ is not isomorphic to $W_{4}$, and $W_{4}^{o}$ can be obtained from $(H,\Omega)$ by deleting or contracting one edge. If $W_{4}^{o}$ can be obtained from $(H,\Omega)$ by deleting an edge $e$, then $e$ connects nonadjacent vertices of $W_{4}^{o}$. In this case, $(H,\Omega)$ has a weak $K_{4}^{o}$- or $K_{4}^{e}$-minor by Lemma 9, and hence $(G,\Sigma)$ has a $K_{4}^{o}$- or $K_{4}^{e}$-minor. If $W_{4}^{o}$ can be obtained from $(H,\Omega)$ by contracting an edge $e$, then $(H,\Omega)$ has a weak $K_{4}^{o}$- or $K_{4}^{e}$-minor by Lemma 10, and hence $(G,\Sigma)$ has a weak $K_{4}^{o}$- or $K_{4}^{e}$-minor. ∎

Since $G$ has no $K_{4}$- and no $K_{2,3}$-minor, $G$ is outerplanar. Hence $G$ can be embedded in the plane such that all its vertices are incident to the infinite face. Add edges such that all finite faces are either triangles or cycles with exactly two edges, and let the resulting graph be $H$. Construct the following tree $R$ . The vertices of $R$ are all finite faces of the plane embedding. Connect two vertices of the tree if the corresponding faces have an edge in common. Then $R$ is a path. For if not, there would be a face that has edges in common with at least three other faces. Such a graph has a $K_{3}^{=}$-minor. By induction it can now be shown that $H$ is a $2$-path. Hence $G$ is a partial $2$-path. ∎

If $G$ has a $W_{4}$-minor, then $(G,\Sigma)$ is isomorphic to $W_{4}^{o}$, by Lemma 11. We may therefore assume that $G$ has no $W_{4}$-minor.

Suppose $G$ has a $K_{4}$-minor. Then, by Lemma 6, $(G,\Sigma)$ has a wide separation $[G_{1},G_{2}]$. For $i=1,2$, let $(H_{i},\Omega_{i})$ be obtained from $(G_{i},E(G_{i})\cap\Sigma)$ by adding between the vertices of attachment of $(G_{i},E(G_{i})\cap\Sigma)$ in the wide separation $[G_{1},G_{2}]$ an odd and even edge in parallel. Then $(H_{i},\Omega_{i})$, $i=1,2$, contains no weak minor isomorphic to $K_{4}^{e}$, $K_{4}^{o}$, $K_{2,3}^{e}$, $K_{3}^{=}$, or $W_{4}^{o}$, for otherwise $(G,\Sigma)$ would contain a weak minor isomorphic to $K_{4}^{e}$, $K_{4}^{o}$, $K_{2,3}^{e}$, $K_{3}^{=}$, or $W_{4}^{o}$.

Let $u$ and $v$ be the vertices of attachment of $(G_{1},E(G_{1})\cap\Sigma)$ in the wide separation $[G_{1}.G_{2}]$. Suppose $G_{1}-\{u,v\}$ contains more than one component; let $C_{1},\ldots,C_{k}$ be the components. Then, as $G$ is $2$-connected, each $G_{1}[V(C_{i})\cup\{u,v\}]$ contains a path of length $\geq 2$ connecting $u$ and $v$. If there are components $C_{j}$ and $C_{k}$ such that both $G_{1}[V(C_{j})\cup\{u,v\}]$ and $G_{1}[V(C_{k})\cup\{u,v\}]$ contain an even path connecting $u$ and $v$, then $(G,\Sigma)$ has a $K_{2,3}^{e}$-minor, a contradiction. Similarly, there are no two components $C_{j}$ and $C_{k}$ such that both $G_{1}[V(C_{j})\cup\{u,v\}]$ and $G_{1}[V(C_{k})\cup\{u,v\}]$ contain an odd path connecting $u$ and $v$. Hence $G_{1}-\{u,v\}$ has exactly two components $C_{1}$ and $C_{2}$. If $G[V(C_{1})\cup\{u,v\}]$ contains a path $P$ of length $\geq 3$ connecting $u$ and $v$ and $G[V(C_{2})\cup\{u,v\}]$ contains an even path connecting $u$ and $v$, then, possibly after re-signing, we contract an edge of $P$ to obtain an even path. Then $(G,\Sigma)$ has a $K_{2,3}^{e}$-minor. The cases where $G[V(C_{2})\cup\{u,v\}]$ has an odd path connecting $u$ and $v$, and where $G[V(C_{2})\cup\{u,v\}]$ has a path of length $\geq 3$ connecting $u$ and $v$ are similar.

If $G[V(C_{1})\cup\{u,v\}]$ has parallel edges whose ends are not $u$ and $v$, then $G[V(C_{1})\cup\{u,v\}]$ and $G[V(C_{2})\cup\{u,v\}]$ contain paths connecting $u$ and $v$ of equal parity. Then $(G,\Sigma)$ has a $K_{2,3}^{e}$. A similar statement holds for $G[V(C_{2})\cup\{u,v\}]$. Hence $G[V(C_{1})\cup\{u,v\}]$ and $G[V(C_{2})\cup\{u,v\}]$ are paths of length $2$ and have different parity. Thus $(H_{1},\Omega_{1})$ is a subgraph of a sided wide $2$-path where one of the parallel edges between $u$ and $v$ is a side.

Suppose $G_{1}-\{u,v\}$ contains exactly one component. By induction $(H_{1},\Omega_{1})$ is a partial wide $2$-path. Since, in the construction of a wide $2$-path, parallel edges appear only parallel to the edges in $\mathcal{F}$ of a sided wide $2$-path and $(H_{1},\Omega_{1})-\{u,v\}$ has exactly one component, there exists a set $\mathcal{F}_{1}$ of two distinct edges, one of which is between $u$ and $v$, of $(H_{1},\Omega_{1})$ such that $[(H_{1},\Omega_{1}),\mathcal{F}_{1}]$ is a sided wide $2$-path.

Similarly there exists a set $\mathcal{F}_{2}$ of two distinct edge, one of which is between the vertices of attachments of $(G_{2},\Sigma_{2})$ such that $[(H_{2},\Omega_{2}),\mathcal{F}_{2}]$ is a sided wide $2$-path. Then $(G,\Sigma)$ is a partial wide $2$-path. We may therefore assume that $G$ has no $K_{4}$-minor.

Suppose $G$ has a $K_{2,3}$-minor. Then, by Lemma 7, $(G,\Sigma)$ has a wide separation $[G_{1},G_{2}]$, where $G_{1}$ is isomorphic to $K_{2}^{c}$. Let $(H_{2},\Omega_{2})$ be obtained from $(G_{2},E(G_{2})\cap\Sigma)$ by adding between the vertices attachment of $(G_{2},E(G_{2})\cap\Sigma)$ in the wide separation $[G_{1},G_{2}]$ an odd and even edge in parallel. Then $(H_{2},\Omega_{2})$ contains no minor isomorphic to $K_{4}^{e}$, $K_{4}^{o}$, $K_{2,3}^{e}$, $K_{3}^{=}$, or $W_{4}^{o}$. By induction $(H_{2},\Omega_{2})$ is a partial wide $2$-path. Similar as above, there is a sided wide $2$-path $[(J_{2},\Delta_{2}),\mathcal{F}_{2}]$ such that $(H_{2},\Omega)$ is a subgraph of $(J_{2},\Delta_{2})$ and one of the edges of $\mathcal{F}_{2}$ is an edge between the attachments of $(G_{2},E(G_{2})\cap\Sigma)$ in the wide separation $[G_{1},G_{2}]$. Then $(G,\Sigma)$ is a partial wide $2$-path. We may therefore assume that $G$ has no $K_{2,3}$-minor.