A young baby and a toddler escaped serious injury following an arson attack in Moygashel, near Dungannon on Saturday.
Detectives have appealed for information relating to the attack on Jacksonville Road, to come forward.
The kitchen of the house was filled with smoke from a fire which was started in a shed at the back of the property after 3am.
Inspector Joanne Gibson said: “At approximately 3.40am NIFRS received a report that a shed was on fire at the back of a house on Jacksonville Road.
“Police attended the scene along with the NIFRS who extinguished the blaze.
“A shed and oil tank were destroyed in this fire. Part of the garden fence was also damaged.
“The male occupant of the house heard one of the windows at the back of his property shatter and when he went to investigate the kitchen was filled with smoke from the fire outside.
“He was able to take his partner and two children (a baby and toddler) to safety.
“Fortunately no one was injured during this incident, however this was an extremely traumatic incident for all concerned.
“This was a cowardly attack on a family who should be able to feel safe in their own home with their young children. No one has the right to take this away from anyone.”
Inspector Gibson added: “I would appeal to anyone who noticed any suspicious activity in this area around the time of the incident to contact detectives at Cookstown on the non-emergency number 101, quoting reference 219 30/11/19.
“Information can also be provided anonymously to the independent charity Crimestoppers on 0800 555 111.”