(K_(1,4);2)-图的闭包和路长

Closure and Length of Longest Path of (K_(1,4); 2)-Graphs

为了推广无爪图G在闭包运算下是唯一确定的并且保持路长不变这一结论,对包含无爪图的(K_(1,4);2)-图进行研究,主要采用逐一讨论、排除的方法对此类图的路长在闭包运算下保持不变的性质进行证明。结果表明:在已知K_1∨P_4-free或T_3-free的(K_(1,4);2)-图在闭包运算下也唯一确定并且仍为(K_(1,4);2)-图的条件下,如果G是K_1∨P_4-free或T_3-free的(K_(1,4);2)-图,则在闭包的运算下保持路长不变;K1∨P4-free或T3-free的(K_(1,4);2)-图G可迹当且仅当其闭包是可迹的,其中K_1∨P_4为一个点与长为4的路的联图,T_3为K_(1,3)与K_2的并图。

To promote the conclusion of the closure of a claw-free graph G being well-definded and the length of a longest path in G or in its closure being the same, the （K1,4 ;2）-graphs containing claw-free graphs were studied. The property of the length of a longest path in this class of graph or in its closure being the same was proved by using the method of discussing each kind of case one by one and rule out one by one. The results show that under the condition of any （K1,4 ;2）- graph being K1 V P4-free or T3-free whose closure is well-definded and is still a （ K1,4 ; 2 ）-graph, if G is a （ K1,4 ; 2 ）- graph which is K1 VP4-free or T3-free, the length of a longest path in Gorin its closure is the same. A （ K1,4 ;2 ）-graph which is K1 VP 4-free or T3-free is traceable if and only if its closure is traceable, where K1 V P4 expresses the join graph of a vertex and a path whose length is four, and T3 expresses the union graph of K1,3 and K2.