  If $E$ is a directed graph, $K$ is a field, and $I$ is a graded ideal of the
Leavitt path algebra $L_K(E),$ $I$ is completely determined by an admissible
pair $(H,S)$ of two sets of vertices of $E$. The ideal $I=I(H,S)$ is graded
isomorphic to the Leavitt path algebra of the {\em porcupine graph} of $(H,S)$
and the quotient $L_K(E)/I$ is graded isomorphic to the Leavitt path algebra of
the {\em quotient graph} of $(H,S).$ We present a construction which
generalizes both constructions and enables one to consider quotients of graded
ideals: if $(H,S)$ and $(G,T)$ are admissible pairs such that $I(H,S)\subseteq
I(G,T)$, we define the {\em porcupine-quotient graph} $(G,T)/(H,S)$ such that
its Leavitt path algebra is graded isomorphic to the quotient $I(G,T)/I(H,S).$
  Using the porcupine-quotient construction, the existence of a graded
composition series of $L_K(E)$ is equivalent to the existence of a finite chain
of admissible pairs of $E,$ starting with the trivial and ending with the
improper pair, such that the quotient of two consecutive pairs is cofinal (a
graph is cofinal exactly when its Leavitt path algebra is graded simple). We
characterize the existence of such a composition series with a set of
conditions which also provides an algorithm for obtaining such a series. The
conditions are presented in terms of four types of vertices which are all
``terminal'' in a certain sense. Three types are often referred to as the three
primary colors and the fourth type is new. As a corollary, a unital Leavitt
path algebra has a graded composition series.
  We show that the existence of a composition series of $E$ is equivalent to
the existence of a composition series of the graph monoid $M_E$ as well as a
composition series of the talented monoid $M_E^\Gamma.$ An ideal of
$M_E^\Gamma$ is minimal exactly when it is generated by the element of
$M_E^\Gamma$ corresponding to a terminal vertex.
