| Summary: | 博士 === 國立交通大學 === 應用數學系 === 87 === The intersection graph of a family ${\cal F}$ of sets is the graph whose vertices have a one-to-one correspondence to the sets in ${\cal F}$, and two distinct vertices are adjacent if and only if their corresponding sets intersect. An interval graph is an intersection graph of intervals on the real line. c
In Chapter 1, we introduce basic terminology and facts about
interval graphs and their generalizations--interval numbers and
total interval numbers. In Chapter 2, we study interval numbers of powers of block graphs. In particular, we prove that the interval number of the $k$th power of a block graph is at most $k+1$. We then characterize block graphs whose $k$th powers are interval graphs. In Chapter 3, we study total interval numbers of complete $r$-partite graphs. We give bounds of the total interval numbers of complete $r$-partite graphs and determine exact values for some cases. In Chapter 4, we study the total interval numbers of block graphs. We design an algorithm for finding the total interval number of a block graph. In Chapter 5, we give simple proofs for families of graphs closed under taking powers. In Chapter 6, we find the complete set of forbidden subgraphs for substar graphs. We design an algorithm to recognize whether a chordal graph is a substar graph.
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