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The study of hypersurface quadrilateral singularities can bereduced to the study of elliptic K3 surfaces with a singularfiber of type I _ 0 (superscript _, subscript 0), andtherefore these notes consider, besides the topics of thetitle, such K3 surfaces too.The combinations of rational double points that can occur onfibers in the semi-universal deformations of quadrilateralsingularities are examined, to show that the possiblecombinations can be described by a certain law from theviewpoint of Dynkin graphs. This is equivalent to sayingthat the possible combinations of singular fibers inelliptic K3 surfaces with a singular fiber of type I _ 0(superscript _, subscript 0) can be described by a certainlaw using classical Dynkin graphs appearing in the theoryof semi-simple Lie groups. Further, a similar descriptionfor thecombination of singularities on plane sextic curvesis given. Standard knowledge of algebraic geometry atthelevel of graduate students is expected. A new method basedon graphs will attract attention of researches.