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The end secured with two dowels will develop a crack in main panel. Since length of edge piece doesn't change having it fixed across width of main panel from expanding.
Alternatively, opposite end of cutting board is only fixed in center. The main panel is free to expand and contract towards ends. The only visual difference would be that ends of edge piece are no longer flush with edge of main panel.
The second example uses a large glued up panel such as a table top. If table top was fixed to rails across width of panel rise and fall of humidity would undoudtable cause panel to split.
The best method to overcome this is use wooden clip to attach two pieces. A 1/4" x 1/4" dado milled on inside upper edge of rails. This accepts a wooden clip that fits into dado and is then secures to underside of table top. This will security hold top hold top onto rails without restricting expansion and contraction of large panel. Without this table top would slowly start to open up.
In general anytime you are fixing one piece of stock across a large glued up panel measures need to be taken to allow panel to move. Remember that it will expand across its width but not its length. Using one of two methods mentioned above you will be able to prevent your glued up panels from splitting.
Dave Markel is the author of "The All Wood Working Journal". He has helped hundreds of individuals improve their wood working skills. Visit his site at http://all-wood-working-plans.com Subscribe to the All Wood Working Journal at http://all-wood-working-plans.com/wood-working-tips.html