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  "body": "\n## Definition\n\nLet $G$ and $H$ be finite [[graphs|0JD5C82QMB]].\nLet $uv$ be an edge of $G$.\nThe contraction $G / uv$ is obtained from $G$ by removing $u$ and $v$ and adding a new vertex $w$.\nThe new vertex $w$ is adjacent to each other vertex that was adjacent to $u$ or to $v$.\nThe graph $H$ is a minor of $G$ if finitely many edge contractions turn a [[subgraph|0P11X5HKAX]] of $G$ into a graph [[isomorphic|00CT0PTH74]] to $H$.\n\n## Remarks\n\nEquivalently, there are pairwise disjoint nonempty sets $B_x \\subseteq V(G)$, one for each vertex $x$ of $H$, with two properties.\nEach $B_x$ induces a [[connected|0FFAYNMAJZ]] subgraph of $G$.\nFor each edge $xy$ of $H$, some edge of $G$ joins $B_x$ to $B_y$.\nThe sets $B_x$ are called branch sets.\nExample: $K_5$ is a minor of the [[Petersen graph|0NR95CTFCZ]].\nTake as branch sets the five spokes, each made of an outer vertex and its inner neighbour.\nThen the outer $5$-cycle and the inner pentagram join every two branch sets.\nNon-example: the triangle $K_3$ is not a minor of any tree, because deleting and contracting edges of a forest always gives a forest.\nEvery subgraph is a minor, but a minor need not be a subgraph.\nFor example, $K_3$ is a minor of every cycle $C_n$ with $n \\ge 3$ but a subgraph only of $C_3$.\n",
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