Tag: 김동규

Tutte (1961) proved that every simple $3$-connected graph $G$ has an edge $e$ such that $G \setminus e$ or $G / e$ is simple $3$-connected, unless $G$ is isomorphic to a wheel. We call such an edge non-essential. Oxley and Wu (2000) proved that every simple $3$-connected graph has at least $2$ non-essential edges unless it is isomorphic to a wheel. Moreover, they proved that every simple $3$-connected graph has at least $3$ non-essential edges if and only if it is isomorphic to neither a twisted wheel nor a $k$-dimensional wheel with $k\geq2$.

We prove analogous results for graphs with vertex-minors. For a vertex $v$ of a graph $G$, let $G*v$ be the graph obtained from $G$ by deleting all edges joining two neighbors of $v$ and adding edges joining non-adjacent pairs of two neighbors of $v$. This operation is called the local complementation at $v$, and we say two graphs are locally equivalent if one can be obtained from the other by applying a sequence of local complementations. A graph $H$ is a vertex-minor of a graph $G$ if $H$ is an induced subgraph of a graph locally equivalent to $G$. A split of a graph is a partition $(A,B)$ of its vertex set such that $|A|,|B| \geq 2$ and for some $A’\subseteq A$ and $B’\subseteq B$, two vertices $x\in A$ and $y\in B$ are adjacent if and only if $x\in A’$ and $y\in B’$. A graph is prime if it has no split.

A vertex $v$ of a graph is non-essential if at least two of three kinds of vertex-minor reductions at $v$ result in prime graphs. We prove that every prime graph with at least $5$ vertices has at least two non-essential vertices unless it is locally equivalent to a cycle. It is stronger than a theorem proved by Allys (1994), which states that every prime graph with at least $5$ vertices has a non-essential vertex unless it is locally equivalent to a cycle. As a corollary of our result, one can obtain the first result of Oxley and Wu. Furthermore, we show that every prime graph with at least $5$ vertices has at least $3$ non-essential vertices if and only if it is not locally equivalent to a graph with two specified vertices $x$ and $y$ consisting of at least two internally-disjoint paths from $x$ to $y$ in which $x$ and $y$ have no common neighbor.

This is joint work with Sang-il Oum.

Bouchet (1987) defined delta-matroids by relaxing the base exchange axiom of matroids. Oum (2009) introduced a graphic delta-matroid from a pair of a graph and its vertex subset. We define a $\Gamma$-graphic delta-matroid for an abelian group $\Gamma$, which generalizes a graphic delta-matroid.

For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose vertices are labelled by elements of $\Gamma$. We prove that a certain collection of edge sets of a $\Gamma$-labelled graph forms a delta-matroid, which we call a $\Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet (1987) to find a maximum weight feasible set in such a delta-matroid. We also prove that a $\Gamma$-graphic delta-matroid is a graphic delta-matroid if and only if it is even. We prove that every $\mathbb{Z}_p^k$-graphic delta matroid is represented by some symmetric matrix over a field of characteristic of order $p^k$, and if every $\Gamma$-graphic delta-matroid is representable over a finite field $\mathbb{F}$, then $\Gamma$ is isomorphic to $\mathbb{Z}_p^k$ and $\mathbb{F}$ is a field of order $p^\ell$ for some prime $p$ and positive integers $k$ and $\ell$.

This is joint work with Duksang Lee and Sang-il Oum.