Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five

In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most $\frac{2}{9}m$ or $\frac{1}{5}m$, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2-connectivity requirement

HIST 311 U.S. History I

Course syllabus for HIST 311 U.S. History I Course description: This course provides an historical examination of the United States from the founding of the colonies through the Civil War with special emphasis on religion, nation building, economic development and modernization, and the sectional conflict

(A) Certain Party

Title along the top of the cover with a light blue background and a checkerboard design through the centerhttps://scholarsjunction.msstate.edu/cht-sheet-music/13899/thumbnail.jp