Chord graphs and interval plots

Originally wanted to put all the papers inside all the documents are not proven, the results found that I was too weak to "cover the face of the bear" then put some conclusions to remember it qaq after what interest, then to supplement the proof

Some definitions of what to do with your own brain.

1. For any one picture, the number of <= color, the maximum number of independent sets <= the minimum group coverage. When the graph is a string, both equations are taken as equals.

2. A graph is a chord chart when and only if it has a perfect elimination sequence.

3. The maximum potential algorithm (MCS) asks for a perfect elimination sequence for a string graph. Order each point from N to 1, set label[i] to indicate I with how many labeled nodes are labeled. So each time you select Label[i] The maximum point is labeled.

The MLOGN algorithm simply maintains a heap. O (N+M) can consider the J-linked list for each label[i]=j to indicate whose label is J.

4. Determine if a sequence is a perfect elimination sequence.

Set V[I+1......N] [v[i] adjacent to the past, followed by v[j1]v[j2] ... V[JK] As long as the v[j1] and other nodes are adjacent. (Obvious correctness)

Complexity O (n+m) We can consider judging from 1 to n, each time I correspond to the J1 linked list of the J1 need to be adjacent to, and then the complexity should be the same as O (n+m)

5. Set N (i) to represent a collection of nodes adjacent to the I node and after I in the perfect elimination sequence. By definition, i∪n (i) is a regiment.

Then the largest regiment must be in the form of i∪n (i). Prove that you can bring out the front of the largest regiment.

Similarly, it can be proved that all the groups are i∪n (i) or a subset of I∪n (i).

6. The maximum point independent set of the chord graph can be selected after the perfect elimination sequence is obtained from the travel. Why ）

7. The Group of I∪n (i) of each point in which the maximum point is independently concentrated after the maximum point independent set is calculated is a minimum group cover. (The definition of n (i) here is the same as the original?) ）

8. Perfect picture refers to a graph of any one of the induced sub-graph to meet the number of numbers = chromatic number, and the perfect picture refers to any one of the induced sub-graph to meet the maximum point independent set = Minimum Group coverage

Perfect picture = companion Perfect picture. The chord chart is the perfect picture.

9. The interval diagram is a chord chart.

10. Given n intervals, it is required to select the most interval so that the intervals do not overlap each other. is actually the maximum point independent set of interval graphs.

11. There are n bricks, height is 1, the width of the first building block is [Li, Ri], select a block of falling order to make the final block total height as small as possible. is actually the minimum number of staining for interval graphs.

12. A perfect elimination sequence of interval graphs can be obtained by sorting all the intervals according to the right endpoint from small to large.

The back of the "cover-face bear."

Chord graphs and interval plots