اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNerves, minors, and piercing numbers
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We make the first step towards a nerve theorem for graphs. Let $G$ be a simple graph and let $mathcal{F}$ be a family of induced subgraphs of $G$ such that the intersection of any members of $mathcal{F}$ is either empty or connected. We show that if the nerve complex of $mathcal{F}$ has non-vanishing homology in dimension three, then $G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $(p,q)$ theorem due to Alon and Kleitman: Let $mathcal{F}$ be a finite family of open connected sets in the plane such that the intersection of any members of $mathcal{F}$ is either empty or connected. If among any $p geq 3$ members of $mathcal{F}$ there are some three that intersect, then there is a set of $C$ points which intersects every member of $mathcal{F}$, where $C$ is a constant depending only on $p$.
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We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Qcup Rcup Scup T)$ such that there are two ways to reduce $G$ by a vertex-minor operatio
n while preserving the connectivity between $Q$ and $R$ and the connectivity between $S$ and $T$. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids.
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