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''Chvátal watchman theorem''

The following question was posed by V. Klee: How many guards are necessary (and sufficient) to guard (visually cover) a polygonal room (an art gallery) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110230/c1102301.png" /> vertices?

The question was answered by V. Chvátal [[#References|[a1]]]. He proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110230/c1102302.png" /> guards are sometimes necessary and always sufficient to guard a polygonal room of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110230/c1102303.png" /> vertices.

A concise proof was later found by S. Fisk [[#References|[a2]]]. See also [[Art gallery theorems|Art gallery theorems]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V. Chvátal,   "A combinatorial theorem in plane geometry"  ''J. Combin. Th. B'' , '''18'''  (1975)  pp. 39–41</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Fisk,   "A short proof of Chvátal's watchman theorem"  ''J. Combin. Th. B'' , '''24'''  (1978)  pp. 374</TD></TR></table>