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Light subgraphs of graphs embedded in the plane. A survey. (English) Zbl 1259.05045
Summary: It is well known that every planar graph contains a vertex of degree at most 5. A theorem of Kotzig states that every 3-connected planar graph contains an edge whose endvertices have degree-sum at most 13. I. Fabrici and S. Jendrol’ [Graphs Comb. 13, No. 3, 245–250 (1997; Zbl 0891.05025)] proved that every 3-connected planar graph $$G$$ that contains a $$k$$-vertex path contains also a $$k$$-vertex path $$P$$ such that every vertex of $$P$$ has degree at most $$5k$$. A result by H. Enomoto and K. Ota [J. Graph Theory 30, No. 3, 191–203 (1999; Zbl 0916.05020)] says that every 3-connected planar graph $$G$$ of order at least $$k$$ contains a connected subgraph $$H$$ of order $$k$$ such that the degree sum of vertices of $$H$$ in $$G$$ is at most $$8k-1$$.
Motivated by these results, a concept of light graphs has been introduced. A graph $$H$$ is said to be light in a family $$\mathcal{G}$$ of graphs if at least one member of $$\mathcal G$$ contains a copy of $$H$$ and there is an integer $$w(H,\mathcal{G})$$ such that each member $$G$$ of $$\mathcal{G}$$ with a copy of $$H$$ also has a copy $$K$$ of $$H$$ such that $$\sum_{v \in V(K)} \deg_{G}(v) \leq w(H,\mathcal{G})$$.
In this paper we present a survey of results on light graphs in different families of plane graphs and multigraphs. A similar survey dealing with the family of all graphs embedded in surfaces other than the sphere was prepared as well.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C22 Signed and weighted graphs
##### Keywords:
light subgraphs; weight of subgraph; plane graphs
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