The vandalism occurred between Wednesday and Thursday as the world commemorated those who died for our freedom 75 years on from the D-Day landings.
Criminals smashed the headstones on several plots at a Commonwealth Graves Commission, which had been provided in remembrance of the 75th anniversary, at Hirst Wood burial ground in Shipley, Bradford.
One of the targeted graves was of Arthur Sheard, a WWI driver from Saltaire who died in May 1918 at Morton Banks Military Hospital in Keighley. Mr Hirst's four-year-old daughter Hilda died the next day of an illness and the two were tragically buried together at the plot at Hirst Wood.
A spokeswoman for the Hirst Wood Regeneration Group said: "Sad to report that a number of gravestones in Hirst Wood Cemetery have been smashed.
"It is mainly the Commonwealth War Grave Commission stones, including that of Arthur Sheard who was buried at the same time as his four year old daughter Hilda, who died two days after him.
"It is impossible to understand the callous thoughtlessness of those who did this. Do they boast to their friends and relations that they managed to smash gravestones? What possible satisfaction can they get from such a mindless act?
"If anyone has information that might help the police, who have been informed, please get in touch with them."
Detective Inspector Amanda Middleton, said: “Our investigation is ongoing into the vandalism of these gravestones. The gravestones were destroyed in what was a mindless act of destruction and I would urge anyone with any information to come forward and speak to the police.
“High visibility reassurance patrols have been stepped up in the area whilst officers continue with their enquiries.”
Anyone who witnessed this incident or has any information is asked to contact Bradford District Police via 101 quoting log number 952 of 6 June. Information can also be passed to independent charity Crimestoppers on 0800 555 111.