If eumelanine is entirely excluded from the coat, as with the e-locus yellow or red dog, then brindle will not express itself, even if present.
However the other dilution genes do not hamper the expression of brindle. The dog will display chocolate (b), blue (d) or faded stripes (ch) in the areas that restrict eumelanine.
In general dominant black supresses brindle. However occasional dominant black individuals will be seen with one or a few bright fawn stripes on their bodies.
The pattern of the stripes in a brindle is very close to the pattern of segmental cutaneous nerve innervation (dermatomes) and my suspicion is that brindling is produced by an inter action between the migrating proto melanocytes in the embryo and the segmental body plan that underlies all vertebrate development
Brindle acts as a dominant in relation to the absence of brindle.
Little assigned brindle to the E locus along with black mask. Wether it belongs on the e-locus I am not sure. This will be difficult to demonstrate because e-locus supresses all production of eumelanine. One would need to be unable to produce a dog that was homozygous for both brindling and extension yellow. However if one could produce a dog homozygous for both brindling and extension yellow and then breed it to an a-y red and get all brindle puppies one would demonstrate that brindle was not on the same locus as extension yellow.
I do have strong evidence in Borzois brindle is inherited independently of black mask. If brindle and black mask were on the same locus then a black masked brindle dog bred to an a-y, unmasked red, would produce only masked ay reds and unmasked brindle puppies. However I have had a number of black masked Borzoi who when bred to a-y reds produced all four color combinations, black masked and unmasked a-y reds, black masked and unmasked brindles. I currently have a male Borzoi, Aatis, phenotypically he is a black masked brindle. He has sired a total of 13 puppies bred to two non-brindle bitches. All of them are brindle. Both of his parents were brindle. His sire was an unmasked brindle, his dam a black masked brindle. All of his littermates were brindle.
Brindle sighthounds generally have multibanded (agoutied hairs). I have seen this directly on Borzoi, Irish Wolfhounds and Scottish Deerhounds.