The detection of an over-represented sub-sequence in a set of (carefully chosen) DNA sequences is often the main clue leading to the investigation of a possible functional role for such a subsequence. Over-represented substrings (with possibly local mutations) in a biological string are termed motifs. A typical functional unit that can be modeled by a motif is a Transcription Factor Binding Site (TFBS), a portion of the DNA sequence apt to the binding of a protein that participates in complex transcriptomic biochemical reactions. In the literature it has been proposed a simplified combinatorial problem called the planted (l-d)-motif problem (known also as the (l-d) Challenge Problem) that captures the essential combinatorial nature of the motif finding problem. In this paper we propose a novel graph-based algorithm for solving a refinement of the (l-d) Challenge Problem. Experimental results show that instances of the (l-d) Challenge Problem considered difficult for competing state of the art methods in literature can be solved efficiently in our framework.