「聯盟」與「中繼集」在圖論是一個相當新穎而且有趣的觀念，其具有多樣變化的應用，而Sierpiński相關類別圖形目前有許多有意義的性質與各種不同觀點的研究。

圖G = (V, E)的「強防禦聯盟」定義為一個點集合S ⊆ V滿足以下條件，在S集合中任一點v ∈ S，它的相鄰點包含自己同屬於S集合的點數，大於不屬於S集合的相鄰點數。若一強防禦聯盟稱為全域，則同時必須也滿足支配集的定義，稱為「全域強防禦聯盟」。在本論文中第一個主題為在Sierpiński相關類別圖形中找出最小的全域強防禦聯盟，並且也證明推導出的精確值為全域強防禦聯盟的最佳解。

圖G = (V, E)的「中繼集」定義為一個點集合Q ⊆ V滿足以下條件，在圖中任一對點u, v ∈ V \ Q，存在一條路徑，使得此路徑由u到v經過的點都在Q集合中。「中繼數」指在圖中最小數目的中繼集。在本論文中第二個主題為在Sierpiński相關類別圖形中找出中繼數，並且也證明推導出的精確值為中繼集的最佳解。

The alliance problem and the hub set problem are relatively new concepts in graph theory and have many interesting and practical variations. Sierpinski-like Graphs have many appealing properties and were studied from different points of view.

A strong alliance in a graph G = (V, E) is a set of vertices S⊆V satisfying the condition that, for each v∈S, the number of its neighbors, including itself, in S is greater than the number of those neighbors not in S. A strong alliance S is global if S forms a dominating set of G. The first subject of this dissertation is to propose a way for finding a minimum global strong alliance for each of those Sierpinski-like graphs. Furthermore, we also derive the exact values of those global strong alliance numbers.

A set Q⊆V is a hub set of a graph G=(V,E) if, for every pair of vertices u,v∈V\ Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. The second subject of this dissertation is to derive the hub numbers of Sierpinski-like graphs including: Sierpinski graphs, extended Sierpinski graphs, and Sierpinski gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained.

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